{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0001","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A=∅、B=∅，函數表為 ∅，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"反函數表=∅。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054001,"parameters":{"A":[],"B":[],"pairs":[],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[],\"B\":[],\"operation\":\"inverse_table\",\"pairs\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0002","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-3, -2, -1}、B={1, 3, 5}，函數表為 {(-3, 5), (-2, 1), (-1, 3)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -2), (3, -1), (5, -3)}","explanation_zh":"反函數表={(1, -2), (3, -1), (5, -3)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054002,"parameters":{"A":[-3,-2,-1],"B":[1,3,5],"pairs":[[-3,5],[-2,1],[-1,3]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-3,-2,-1],\"B\":[1,3,5],\"operation\":\"inverse_table\",\"pairs\":[[-3,5],[-2,1],[-1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0003","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-4, -3, -2, -1}、B={1, 3, 5, 7}，函數表為 {(-4, 7), (-3, 1), (-2, 3), (-1, 5)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -3), (3, -2), (5, -1), (7, -4)}","explanation_zh":"反函數表={(1, -3), (3, -2), (5, -1), (7, -4)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054003,"parameters":{"A":[-4,-3,-2,-1],"B":[1,3,5,7],"pairs":[[-4,7],[-3,1],[-2,3],[-1,5]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-4,-3,-2,-1],\"B\":[1,3,5,7],\"operation\":\"inverse_table\",\"pairs\":[[-4,7],[-3,1],[-2,3],[-1,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0004","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9}，函數表為 {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -4), (3, -3), (5, -2), (7, -1), (9, -5)}","explanation_zh":"反函數表={(1, -4), (3, -3), (5, -2), (7, -1), (9, -5)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054004,"parameters":{"A":[-5,-4,-3,-2,-1],"B":[1,3,5,7,9],"pairs":[[-5,9],[-4,1],[-3,3],[-2,5],[-1,7]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9],\"operation\":\"inverse_table\",\"pairs\":[[-5,9],[-4,1],[-3,3],[-2,5],[-1,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0005","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11}，函數表為 {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -6)}","explanation_zh":"反函數表={(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -6)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054005,"parameters":{"A":[-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11],"pairs":[[-6,11],[-5,1],[-4,3],[-3,5],[-2,7],[-1,9]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11],\"operation\":\"inverse_table\",\"pairs\":[[-6,11],[-5,1],[-4,3],[-3,5],[-2,7],[-1,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0006","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13}，函數表為 {(-7, 13), (-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -6), (3, -5), (5, -4), (7, -3), (9, -2), (11, -1), (13, -7)}","explanation_zh":"反函數表={(1, -6), (3, -5), (5, -4), (7, -3), (9, -2), (11, -1), (13, -7)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054006,"parameters":{"A":[-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13],"pairs":[[-7,13],[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13],\"operation\":\"inverse_table\",\"pairs\":[[-7,13],[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0007","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15}，函數表為 {(-8, 15), (-7, 1), (-6, 3), (-5, 5), (-4, 7), (-3, 9), (-2, 11), (-1, 13)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -8)}","explanation_zh":"反函數表={(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -8)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054007,"parameters":{"A":[-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15],"pairs":[[-8,15],[-7,1],[-6,3],[-5,5],[-4,7],[-3,9],[-2,11],[-1,13]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15],\"operation\":\"inverse_table\",\"pairs\":[[-8,15],[-7,1],[-6,3],[-5,5],[-4,7],[-3,9],[-2,11],[-1,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0008","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17}，函數表為 {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -8), (3, -7), (5, -6), (7, -5), (9, -4), (11, -3), (13, -2), (15, -1), (17, -9)}","explanation_zh":"反函數表={(1, -8), (3, -7), (5, -6), (7, -5), (9, -4), (11, -3), (13, -2), (15, -1), (17, -9)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054008,"parameters":{"A":[-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17],"pairs":[[-9,17],[-8,1],[-7,3],[-6,5],[-5,7],[-4,9],[-3,11],[-2,13],[-1,15]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17],\"operation\":\"inverse_table\",\"pairs\":[[-9,17],[-8,1],[-7,3],[-6,5],[-5,7],[-4,9],[-3,11],[-2,13],[-1,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0009","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}，函數表為 {(-10, 19), (-9, 1), (-8, 3), (-7, 5), (-6, 7), (-5, 9), (-4, 11), (-3, 13), (-2, 15), (-1, 17)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -10)}","explanation_zh":"反函數表={(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -10)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054009,"parameters":{"A":[-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19],"pairs":[[-10,19],[-9,1],[-8,3],[-7,5],[-6,7],[-5,9],[-4,11],[-3,13],[-2,15],[-1,17]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19],\"operation\":\"inverse_table\",\"pairs\":[[-10,19],[-9,1],[-8,3],[-7,5],[-6,7],[-5,9],[-4,11],[-3,13],[-2,15],[-1,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0010","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}，函數表為 {(-11, 21), (-10, 1), (-9, 3), (-8, 5), (-7, 7), (-6, 9), (-5, 11), (-4, 13), (-3, 15), (-2, 17), (-1, 19)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -10), (3, -9), (5, -8), (7, -7), (9, -6), (11, -5), (13, -4), (15, -3), (17, -2), (19, -1), (21, -11)}","explanation_zh":"反函數表={(1, -10), (3, -9), (5, -8), (7, -7), (9, -6), (11, -5), (13, -4), (15, -3), (17, -2), (19, -1), (21, -11)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054010,"parameters":{"A":[-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21],"pairs":[[-11,21],[-10,1],[-9,3],[-8,5],[-7,7],[-6,9],[-5,11],[-4,13],[-3,15],[-2,17],[-1,19]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21],\"operation\":\"inverse_table\",\"pairs\":[[-11,21],[-10,1],[-9,3],[-8,5],[-7,7],[-6,9],[-5,11],[-4,13],[-3,15],[-2,17],[-1,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0011","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23}，函數表為 {(-12, 23), (-11, 1), (-10, 3), (-9, 5), (-8, 7), (-7, 9), (-6, 11), (-5, 13), (-4, 15), (-3, 17), (-2, 19), (-1, 21)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -12)}","explanation_zh":"反函數表={(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -12)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054011,"parameters":{"A":[-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23],"pairs":[[-12,23],[-11,1],[-10,3],[-9,5],[-8,7],[-7,9],[-6,11],[-5,13],[-4,15],[-3,17],[-2,19],[-1,21]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23],\"operation\":\"inverse_table\",\"pairs\":[[-12,23],[-11,1],[-10,3],[-9,5],[-8,7],[-7,9],[-6,11],[-5,13],[-4,15],[-3,17],[-2,19],[-1,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0012","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}，函數表為 {(-13, 25), (-12, 1), (-11, 3), (-10, 5), (-9, 7), (-8, 9), (-7, 11), (-6, 13), (-5, 15), (-4, 17), (-3, 19), (-2, 21), (-1, 23)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -12), (3, -11), (5, -10), (7, -9), (9, -8), (11, -7), (13, -6), (15, -5), (17, -4), (19, -3), (21, -2), (23, -1), (25, -13)}","explanation_zh":"反函數表={(1, -12), (3, -11), (5, -10), (7, -9), (9, -8), (11, -7), (13, -6), (15, -5), (17, -4), (19, -3), (21, -2), (23, -1), (25, -13)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054012,"parameters":{"A":[-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25],"pairs":[[-13,25],[-12,1],[-11,3],[-10,5],[-9,7],[-8,9],[-7,11],[-6,13],[-5,15],[-4,17],[-3,19],[-2,21],[-1,23]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25],\"operation\":\"inverse_table\",\"pairs\":[[-13,25],[-12,1],[-11,3],[-10,5],[-9,7],[-8,9],[-7,11],[-6,13],[-5,15],[-4,17],[-3,19],[-2,21],[-1,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0013","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27}，函數表為 {(-14, 27), (-13, 1), (-12, 3), (-11, 5), (-10, 7), (-9, 9), (-8, 11), (-7, 13), (-6, 15), (-5, 17), (-4, 19), (-3, 21), (-2, 23), (-1, 25)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -13), (3, -12), (5, -11), (7, -10), (9, -9), (11, -8), (13, -7), (15, -6), (17, -5), (19, -4), (21, -3), (23, -2), (25, -1), (27, -14)}","explanation_zh":"反函數表={(1, -13), (3, -12), (5, -11), (7, -10), (9, -9), (11, -8), (13, -7), (15, -6), (17, -5), (19, -4), (21, -3), (23, -2), (25, -1), (27, -14)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054013,"parameters":{"A":[-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27],"pairs":[[-14,27],[-13,1],[-12,3],[-11,5],[-10,7],[-9,9],[-8,11],[-7,13],[-6,15],[-5,17],[-4,19],[-3,21],[-2,23],[-1,25]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27],\"operation\":\"inverse_table\",\"pairs\":[[-14,27],[-13,1],[-12,3],[-11,5],[-10,7],[-9,9],[-8,11],[-7,13],[-6,15],[-5,17],[-4,19],[-3,21],[-2,23],[-1,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0014","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}，函數表為 {(-15, 29), (-14, 1), (-13, 3), (-12, 5), (-11, 7), (-10, 9), (-9, 11), (-8, 13), (-7, 15), (-6, 17), (-5, 19), (-4, 21), (-3, 23), (-2, 25), (-1, 27)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -14), (3, -13), (5, -12), (7, -11), (9, -10), (11, -9), (13, -8), (15, -7), (17, -6), (19, -5), (21, -4), (23, -3), (25, -2), (27, -1), (29, -15)}","explanation_zh":"反函數表={(1, -14), (3, -13), (5, -12), (7, -11), (9, -10), (11, -9), (13, -8), (15, -7), (17, -6), (19, -5), (21, -4), (23, -3), (25, -2), (27, -1), (29, -15)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054014,"parameters":{"A":[-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29],"pairs":[[-15,29],[-14,1],[-13,3],[-12,5],[-11,7],[-10,9],[-9,11],[-8,13],[-7,15],[-6,17],[-5,19],[-4,21],[-3,23],[-2,25],[-1,27]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29],\"operation\":\"inverse_table\",\"pairs\":[[-15,29],[-14,1],[-13,3],[-12,5],[-11,7],[-10,9],[-9,11],[-8,13],[-7,15],[-6,17],[-5,19],[-4,21],[-3,23],[-2,25],[-1,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0015","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31}，函數表為 {(-16, 31), (-15, 1), (-14, 3), (-13, 5), (-12, 7), (-11, 9), (-10, 11), (-9, 13), (-8, 15), (-7, 17), (-6, 19), (-5, 21), (-4, 23), (-3, 25), (-2, 27), (-1, 29)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -16)}","explanation_zh":"反函數表={(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -16)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054015,"parameters":{"A":[-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31],"pairs":[[-16,31],[-15,1],[-14,3],[-13,5],[-12,7],[-11,9],[-10,11],[-9,13],[-8,15],[-7,17],[-6,19],[-5,21],[-4,23],[-3,25],[-2,27],[-1,29]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31],\"operation\":\"inverse_table\",\"pairs\":[[-16,31],[-15,1],[-14,3],[-13,5],[-12,7],[-11,9],[-10,11],[-9,13],[-8,15],[-7,17],[-6,19],[-5,21],[-4,23],[-3,25],[-2,27],[-1,29]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0016","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}，函數表為 {(-17, 33), (-16, 1), (-15, 3), (-14, 5), (-13, 7), (-12, 9), (-11, 11), (-10, 13), (-9, 15), (-8, 17), (-7, 19), (-6, 21), (-5, 23), (-4, 25), (-3, 27), (-2, 29), (-1, 31)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -16), (3, -15), (5, -14), (7, -13), (9, -12), (11, -11), (13, -10), (15, -9), (17, -8), (19, -7), (21, -6), (23, -5), (25, -4), (27, -3), (29, -2), (31, -1), (33, -17)}","explanation_zh":"反函數表={(1, -16), (3, -15), (5, -14), (7, -13), (9, -12), (11, -11), (13, -10), (15, -9), (17, -8), (19, -7), (21, -6), (23, -5), (25, -4), (27, -3), (29, -2), (31, -1), (33, -17)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054016,"parameters":{"A":[-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33],"pairs":[[-17,33],[-16,1],[-15,3],[-14,5],[-13,7],[-12,9],[-11,11],[-10,13],[-9,15],[-8,17],[-7,19],[-6,21],[-5,23],[-4,25],[-3,27],[-2,29],[-1,31]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33],\"operation\":\"inverse_table\",\"pairs\":[[-17,33],[-16,1],[-15,3],[-14,5],[-13,7],[-12,9],[-11,11],[-10,13],[-9,15],[-8,17],[-7,19],[-6,21],[-5,23],[-4,25],[-3,27],[-2,29],[-1,31]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0017","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35}，函數表為 {(-18, 35), (-17, 1), (-16, 3), (-15, 5), (-14, 7), (-13, 9), (-12, 11), (-11, 13), (-10, 15), (-9, 17), (-8, 19), (-7, 21), (-6, 23), (-5, 25), (-4, 27), (-3, 29), (-2, 31), (-1, 33)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -17), (3, -16), (5, -15), (7, -14), (9, -13), (11, -12), (13, -11), (15, -10), (17, -9), (19, -8), (21, -7), (23, -6), (25, -5), (27, -4), (29, -3), (31, -2), (33, -1), (35, -18)}","explanation_zh":"反函數表={(1, -17), (3, -16), (5, -15), (7, -14), (9, -13), (11, -12), (13, -11), (15, -10), (17, -9), (19, -8), (21, -7), (23, -6), (25, -5), (27, -4), (29, -3), (31, -2), (33, -1), (35, -18)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054017,"parameters":{"A":[-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35],"pairs":[[-18,35],[-17,1],[-16,3],[-15,5],[-14,7],[-13,9],[-12,11],[-11,13],[-10,15],[-9,17],[-8,19],[-7,21],[-6,23],[-5,25],[-4,27],[-3,29],[-2,31],[-1,33]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35],\"operation\":\"inverse_table\",\"pairs\":[[-18,35],[-17,1],[-16,3],[-15,5],[-14,7],[-13,9],[-12,11],[-11,13],[-10,15],[-9,17],[-8,19],[-7,21],[-6,23],[-5,25],[-4,27],[-3,29],[-2,31],[-1,33]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0018","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37}，函數表為 {(-19, 37), (-18, 1), (-17, 3), (-16, 5), (-15, 7), (-14, 9), (-13, 11), (-12, 13), (-11, 15), (-10, 17), (-9, 19), (-8, 21), (-7, 23), (-6, 25), (-5, 27), (-4, 29), (-3, 31), (-2, 33), (-1, 35)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -18), (3, -17), (5, -16), (7, -15), (9, -14), (11, -13), (13, -12), (15, -11), (17, -10), (19, -9), (21, -8), (23, -7), (25, -6), (27, -5), (29, -4), (31, -3), (33, -2), (35, -1), (37, -19)}","explanation_zh":"反函數表={(1, -18), (3, -17), (5, -16), (7, -15), (9, -14), (11, -13), (13, -12), (15, -11), (17, -10), (19, -9), (21, -8), (23, -7), (25, -6), (27, -5), (29, -4), (31, -3), (33, -2), (35, -1), (37, -19)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054018,"parameters":{"A":[-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37],"pairs":[[-19,37],[-18,1],[-17,3],[-16,5],[-15,7],[-14,9],[-13,11],[-12,13],[-11,15],[-10,17],[-9,19],[-8,21],[-7,23],[-6,25],[-5,27],[-4,29],[-3,31],[-2,33],[-1,35]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37],\"operation\":\"inverse_table\",\"pairs\":[[-19,37],[-18,1],[-17,3],[-16,5],[-15,7],[-14,9],[-13,11],[-12,13],[-11,15],[-10,17],[-9,19],[-8,21],[-7,23],[-6,25],[-5,27],[-4,29],[-3,31],[-2,33],[-1,35]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0019","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39}，函數表為 {(-20, 39), (-19, 1), (-18, 3), (-17, 5), (-16, 7), (-15, 9), (-14, 11), (-13, 13), (-12, 15), (-11, 17), (-10, 19), (-9, 21), (-8, 23), (-7, 25), (-6, 27), (-5, 29), (-4, 31), (-3, 33), (-2, 35), (-1, 37)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -19), (3, -18), (5, -17), (7, -16), (9, -15), (11, -14), (13, -13), (15, -12), (17, -11), (19, -10), (21, -9), (23, -8), (25, -7), (27, -6), (29, -5), (31, -4), (33, -3), (35, -2), (37, -1), (39, -20)}","explanation_zh":"反函數表={(1, -19), (3, -18), (5, -17), (7, -16), (9, -15), (11, -14), (13, -13), (15, -12), (17, -11), (19, -10), (21, -9), (23, -8), (25, -7), (27, -6), (29, -5), (31, -4), (33, -3), (35, -2), (37, -1), (39, -20)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054019,"parameters":{"A":[-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39],"pairs":[[-20,39],[-19,1],[-18,3],[-17,5],[-16,7],[-15,9],[-14,11],[-13,13],[-12,15],[-11,17],[-10,19],[-9,21],[-8,23],[-7,25],[-6,27],[-5,29],[-4,31],[-3,33],[-2,35],[-1,37]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39],\"operation\":\"inverse_table\",\"pairs\":[[-20,39],[-19,1],[-18,3],[-17,5],[-16,7],[-15,9],[-14,11],[-13,13],[-12,15],[-11,17],[-10,19],[-9,21],[-8,23],[-7,25],[-6,27],[-5,29],[-4,31],[-3,33],[-2,35],[-1,37]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0020","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41}，函數表為 {(-21, 41), (-20, 1), (-19, 3), (-18, 5), (-17, 7), (-16, 9), (-15, 11), (-14, 13), (-13, 15), (-12, 17), (-11, 19), (-10, 21), (-9, 23), (-8, 25), (-7, 27), (-6, 29), (-5, 31), (-4, 33), (-3, 35), (-2, 37), (-1, 39)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -20), (3, -19), (5, -18), (7, -17), (9, -16), (11, -15), (13, -14), (15, -13), (17, -12), (19, -11), (21, -10), (23, -9), (25, -8), (27, -7), (29, -6), (31, -5), (33, -4), (35, -3), (37, -2), (39, -1), (41, -21)}","explanation_zh":"反函數表={(1, -20), (3, -19), (5, -18), (7, -17), (9, -16), (11, -15), (13, -14), (15, -13), (17, -12), (19, -11), (21, -10), (23, -9), (25, -8), (27, -7), (29, -6), (31, -5), (33, -4), (35, -3), (37, -2), (39, -1), (41, -21)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054020,"parameters":{"A":[-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41],"pairs":[[-21,41],[-20,1],[-19,3],[-18,5],[-17,7],[-16,9],[-15,11],[-14,13],[-13,15],[-12,17],[-11,19],[-10,21],[-9,23],[-8,25],[-7,27],[-6,29],[-5,31],[-4,33],[-3,35],[-2,37],[-1,39]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41],\"operation\":\"inverse_table\",\"pairs\":[[-21,41],[-20,1],[-19,3],[-18,5],[-17,7],[-16,9],[-15,11],[-14,13],[-13,15],[-12,17],[-11,19],[-10,21],[-9,23],[-8,25],[-7,27],[-6,29],[-5,31],[-4,33],[-3,35],[-2,37],[-1,39]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0021","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43}，函數表為 {(-22, 43), (-21, 1), (-20, 3), (-19, 5), (-18, 7), (-17, 9), (-16, 11), (-15, 13), (-14, 15), (-13, 17), (-12, 19), (-11, 21), (-10, 23), (-9, 25), (-8, 27), (-7, 29), (-6, 31), (-5, 33), (-4, 35), (-3, 37), (-2, 39), (-1, 41)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -21), (3, -20), (5, -19), (7, -18), (9, -17), (11, -16), (13, -15), (15, -14), (17, -13), (19, -12), (21, -11), (23, -10), (25, -9), (27, -8), (29, -7), (31, -6), (33, -5), (35, -4), (37, -3), (39, -2), (41, -1), (43, -22)}","explanation_zh":"反函數表={(1, -21), (3, -20), (5, -19), (7, -18), (9, -17), (11, -16), (13, -15), (15, -14), (17, -13), (19, -12), (21, -11), (23, -10), (25, -9), (27, -8), (29, -7), (31, -6), (33, -5), (35, -4), (37, -3), (39, -2), (41, -1), (43, -22)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054021,"parameters":{"A":[-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43],"pairs":[[-22,43],[-21,1],[-20,3],[-19,5],[-18,7],[-17,9],[-16,11],[-15,13],[-14,15],[-13,17],[-12,19],[-11,21],[-10,23],[-9,25],[-8,27],[-7,29],[-6,31],[-5,33],[-4,35],[-3,37],[-2,39],[-1,41]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43],\"operation\":\"inverse_table\",\"pairs\":[[-22,43],[-21,1],[-20,3],[-19,5],[-18,7],[-17,9],[-16,11],[-15,13],[-14,15],[-13,17],[-12,19],[-11,21],[-10,23],[-9,25],[-8,27],[-7,29],[-6,31],[-5,33],[-4,35],[-3,37],[-2,39],[-1,41]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0022","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45}，函數表為 {(-23, 45), (-22, 1), (-21, 3), (-20, 5), (-19, 7), (-18, 9), (-17, 11), (-16, 13), (-15, 15), (-14, 17), (-13, 19), (-12, 21), (-11, 23), (-10, 25), (-9, 27), (-8, 29), (-7, 31), (-6, 33), (-5, 35), (-4, 37), (-3, 39), (-2, 41), (-1, 43)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -22), (3, -21), (5, -20), (7, -19), (9, -18), (11, -17), (13, -16), (15, -15), (17, -14), (19, -13), (21, -12), (23, -11), (25, -10), (27, -9), (29, -8), (31, -7), (33, -6), (35, -5), (37, -4), (39, -3), (41, -2), (43, -1), (45, -23)}","explanation_zh":"反函數表={(1, -22), (3, -21), (5, -20), (7, -19), (9, -18), (11, -17), (13, -16), (15, -15), (17, -14), (19, -13), (21, -12), (23, -11), (25, -10), (27, -9), (29, -8), (31, -7), (33, -6), (35, -5), (37, -4), (39, -3), (41, -2), (43, -1), (45, -23)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054022,"parameters":{"A":[-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45],"pairs":[[-23,45],[-22,1],[-21,3],[-20,5],[-19,7],[-18,9],[-17,11],[-16,13],[-15,15],[-14,17],[-13,19],[-12,21],[-11,23],[-10,25],[-9,27],[-8,29],[-7,31],[-6,33],[-5,35],[-4,37],[-3,39],[-2,41],[-1,43]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45],\"operation\":\"inverse_table\",\"pairs\":[[-23,45],[-22,1],[-21,3],[-20,5],[-19,7],[-18,9],[-17,11],[-16,13],[-15,15],[-14,17],[-13,19],[-12,21],[-11,23],[-10,25],[-9,27],[-8,29],[-7,31],[-6,33],[-5,35],[-4,37],[-3,39],[-2,41],[-1,43]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0023","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47}，函數表為 {(-24, 47), (-23, 1), (-22, 3), (-21, 5), (-20, 7), (-19, 9), (-18, 11), (-17, 13), (-16, 15), (-15, 17), (-14, 19), (-13, 21), (-12, 23), (-11, 25), (-10, 27), (-9, 29), (-8, 31), (-7, 33), (-6, 35), (-5, 37), (-4, 39), (-3, 41), (-2, 43), (-1, 45)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -23), (3, -22), (5, -21), (7, -20), (9, -19), (11, -18), (13, -17), (15, -16), (17, -15), (19, -14), (21, -13), (23, -12), (25, -11), (27, -10), (29, -9), (31, -8), (33, -7), (35, -6), (37, -5), (39, -4), (41, -3), (43, -2), (45, -1), (47, -24)}","explanation_zh":"反函數表={(1, -23), (3, -22), (5, -21), (7, -20), (9, -19), (11, -18), (13, -17), (15, -16), (17, -15), (19, -14), (21, -13), (23, -12), (25, -11), (27, -10), (29, -9), (31, -8), (33, -7), (35, -6), (37, -5), (39, -4), (41, -3), (43, -2), (45, -1), (47, -24)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054023,"parameters":{"A":[-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47],"pairs":[[-24,47],[-23,1],[-22,3],[-21,5],[-20,7],[-19,9],[-18,11],[-17,13],[-16,15],[-15,17],[-14,19],[-13,21],[-12,23],[-11,25],[-10,27],[-9,29],[-8,31],[-7,33],[-6,35],[-5,37],[-4,39],[-3,41],[-2,43],[-1,45]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47],\"operation\":\"inverse_table\",\"pairs\":[[-24,47],[-23,1],[-22,3],[-21,5],[-20,7],[-19,9],[-18,11],[-17,13],[-16,15],[-15,17],[-14,19],[-13,21],[-12,23],[-11,25],[-10,27],[-9,29],[-8,31],[-7,33],[-6,35],[-5,37],[-4,39],[-3,41],[-2,43],[-1,45]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0024","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49}，函數表為 {(-25, 49), (-24, 1), (-23, 3), (-22, 5), (-21, 7), (-20, 9), (-19, 11), (-18, 13), (-17, 15), (-16, 17), (-15, 19), (-14, 21), (-13, 23), (-12, 25), (-11, 27), (-10, 29), (-9, 31), (-8, 33), (-7, 35), (-6, 37), (-5, 39), (-4, 41), (-3, 43), (-2, 45), (-1, 47)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -24), (3, -23), (5, -22), (7, -21), (9, -20), (11, -19), (13, -18), (15, -17), (17, -16), (19, -15), (21, -14), (23, -13), (25, -12), (27, -11), (29, -10), (31, -9), (33, -8), (35, -7), (37, -6), (39, -5), (41, -4), (43, -3), (45, -2), (47, -1), (49, -25)}","explanation_zh":"反函數表={(1, -24), (3, -23), (5, -22), (7, -21), (9, -20), (11, -19), (13, -18), (15, -17), (17, -16), (19, -15), (21, -14), (23, -13), (25, -12), (27, -11), (29, -10), (31, -9), (33, -8), (35, -7), (37, -6), (39, -5), (41, -4), (43, -3), (45, -2), (47, -1), (49, -25)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054024,"parameters":{"A":[-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49],"pairs":[[-25,49],[-24,1],[-23,3],[-22,5],[-21,7],[-20,9],[-19,11],[-18,13],[-17,15],[-16,17],[-15,19],[-14,21],[-13,23],[-12,25],[-11,27],[-10,29],[-9,31],[-8,33],[-7,35],[-6,37],[-5,39],[-4,41],[-3,43],[-2,45],[-1,47]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49],\"operation\":\"inverse_table\",\"pairs\":[[-25,49],[-24,1],[-23,3],[-22,5],[-21,7],[-20,9],[-19,11],[-18,13],[-17,15],[-16,17],[-15,19],[-14,21],[-13,23],[-12,25],[-11,27],[-10,29],[-9,31],[-8,33],[-7,35],[-6,37],[-5,39],[-4,41],[-3,43],[-2,45],[-1,47]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0025","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"有限雙射的反函數表","stem_zh":"明示總函數 f:A→B，A={-26, -25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51}，函數表為 {(-26, 51), (-25, 1), (-24, 3), (-23, 5), (-22, 7), (-21, 9), (-20, 11), (-19, 13), (-18, 15), (-17, 17), (-16, 19), (-15, 21), (-14, 23), (-13, 25), (-12, 27), (-11, 29), (-10, 31), (-9, 33), (-8, 35), (-7, 37), (-6, 39), (-5, 41), (-4, 43), (-3, 45), (-2, 47), (-1, 49)}，且 f 已確認為雙射。求反函數 g:B→A 的完整有序對集合。","answer":{"kind":"expression","value":"{(1, -25), (3, -24), (5, -23), (7, -22), (9, -21), (11, -20), (13, -19), (15, -18), (17, -17), (19, -16), (21, -15), (23, -14), (25, -13), (27, -12), (29, -11), (31, -10), (33, -9), (35, -8), (37, -7), (39, -6), (41, -5), (43, -4), (45, -3), (47, -2), (49, -1), (51, -26)}","explanation_zh":"反函數表={(1, -25), (3, -24), (5, -23), (7, -22), (9, -21), (11, -20), (13, -19), (15, -18), (17, -17), (19, -16), (21, -15), (23, -14), (25, -13), (27, -12), (29, -11), (31, -10), (33, -9), (35, -8), (37, -7), (39, -6), (41, -5), (43, -4), (45, -3), (47, -2), (49, -1), (51, -26)}。每個目標都有唯一原像，交換配對才得到總函數 B→A；空雙射的反函數也是空函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限雙射的反函數表"]},"variant":{"family_id":"qf-inverse-function-table","seed":202610054025,"parameters":{"A":[-26,-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51],"pairs":[[-26,51],[-25,1],[-24,3],[-23,5],[-22,7],[-21,9],[-20,11],[-19,13],[-18,15],[-17,17],[-16,19],[-15,21],[-14,23],[-13,25],[-12,27],[-11,29],[-10,31],[-9,33],[-8,35],[-7,37],[-6,39],[-5,41],[-4,43],[-3,45],[-2,47],[-1,49]],"operation":"inverse_table","semantic_template":"qf-inverse-function-table/{\"A\":[-26,-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51],\"operation\":\"inverse_table\",\"pairs\":[[-26,51],[-25,1],[-24,3],[-23,5],[-22,7],[-21,9],[-20,11],[-19,13],[-18,15],[-17,17],[-16,19],[-15,21],[-14,23],[-13,25],[-12,27],[-11,29],[-10,31],[-9,33],[-8,35],[-7,37],[-6,39],[-5,41],[-4,43],[-3,45],[-2,47],[-1,49]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0026","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-2, -1}、B={1, 3}，函數表為 {(-2, 3), (-1, 1)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(1)，其中 1∈B。","answer":{"kind":"short_text","value":"-1","explanation_zh":"g(1)=-1。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054026,"parameters":{"A":[-2,-1],"B":[1,3],"pairs":[[-2,3],[-1,1]],"t":1,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-2,-1],\"B\":[1,3],\"operation\":\"inverse_evaluate\",\"pairs\":[[-2,3],[-1,1]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0027","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-3, -2, -1}、B={1, 3, 5}，函數表為 {(-3, 5), (-2, 1), (-1, 3)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(5)，其中 5∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(5)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054027,"parameters":{"A":[-3,-2,-1],"B":[1,3,5],"pairs":[[-3,5],[-2,1],[-1,3]],"t":5,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-3,-2,-1],\"B\":[1,3,5],\"operation\":\"inverse_evaluate\",\"pairs\":[[-3,5],[-2,1],[-1,3]],\"t\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0028","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-4, -3, -2, -1}、B={1, 3, 5, 7}，函數表為 {(-4, 7), (-3, 1), (-2, 3), (-1, 5)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(1)，其中 1∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(1)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054028,"parameters":{"A":[-4,-3,-2,-1],"B":[1,3,5,7],"pairs":[[-4,7],[-3,1],[-2,3],[-1,5]],"t":1,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-4,-3,-2,-1],\"B\":[1,3,5,7],\"operation\":\"inverse_evaluate\",\"pairs\":[[-4,7],[-3,1],[-2,3],[-1,5]],\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0029","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9}，函數表為 {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(3)，其中 3∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(3)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054029,"parameters":{"A":[-5,-4,-3,-2,-1],"B":[1,3,5,7,9],"pairs":[[-5,9],[-4,1],[-3,3],[-2,5],[-1,7]],"t":3,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9],\"operation\":\"inverse_evaluate\",\"pairs\":[[-5,9],[-4,1],[-3,3],[-2,5],[-1,7]],\"t\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0030","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11}，函數表為 {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(5)，其中 5∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(5)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054030,"parameters":{"A":[-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11],"pairs":[[-6,11],[-5,1],[-4,3],[-3,5],[-2,7],[-1,9]],"t":5,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11],\"operation\":\"inverse_evaluate\",\"pairs\":[[-6,11],[-5,1],[-4,3],[-3,5],[-2,7],[-1,9]],\"t\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0031","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13}，函數表為 {(-7, 13), (-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(7)，其中 7∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(7)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054031,"parameters":{"A":[-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13],"pairs":[[-7,13],[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11]],"t":7,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13],\"operation\":\"inverse_evaluate\",\"pairs\":[[-7,13],[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11]],\"t\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0032","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15}，函數表為 {(-8, 15), (-7, 1), (-6, 3), (-5, 5), (-4, 7), (-3, 9), (-2, 11), (-1, 13)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(9)，其中 9∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(9)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054032,"parameters":{"A":[-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15],"pairs":[[-8,15],[-7,1],[-6,3],[-5,5],[-4,7],[-3,9],[-2,11],[-1,13]],"t":9,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15],\"operation\":\"inverse_evaluate\",\"pairs\":[[-8,15],[-7,1],[-6,3],[-5,5],[-4,7],[-3,9],[-2,11],[-1,13]],\"t\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0033","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17}，函數表為 {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(11)，其中 11∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(11)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054033,"parameters":{"A":[-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17],"pairs":[[-9,17],[-8,1],[-7,3],[-6,5],[-5,7],[-4,9],[-3,11],[-2,13],[-1,15]],"t":11,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17],\"operation\":\"inverse_evaluate\",\"pairs\":[[-9,17],[-8,1],[-7,3],[-6,5],[-5,7],[-4,9],[-3,11],[-2,13],[-1,15]],\"t\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0034","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}，函數表為 {(-10, 19), (-9, 1), (-8, 3), (-7, 5), (-6, 7), (-5, 9), (-4, 11), (-3, 13), (-2, 15), (-1, 17)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(13)，其中 13∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(13)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054034,"parameters":{"A":[-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19],"pairs":[[-10,19],[-9,1],[-8,3],[-7,5],[-6,7],[-5,9],[-4,11],[-3,13],[-2,15],[-1,17]],"t":13,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19],\"operation\":\"inverse_evaluate\",\"pairs\":[[-10,19],[-9,1],[-8,3],[-7,5],[-6,7],[-5,9],[-4,11],[-3,13],[-2,15],[-1,17]],\"t\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0035","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}，函數表為 {(-11, 21), (-10, 1), (-9, 3), (-8, 5), (-7, 7), (-6, 9), (-5, 11), (-4, 13), (-3, 15), (-2, 17), (-1, 19)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(15)，其中 15∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(15)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054035,"parameters":{"A":[-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21],"pairs":[[-11,21],[-10,1],[-9,3],[-8,5],[-7,7],[-6,9],[-5,11],[-4,13],[-3,15],[-2,17],[-1,19]],"t":15,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21],\"operation\":\"inverse_evaluate\",\"pairs\":[[-11,21],[-10,1],[-9,3],[-8,5],[-7,7],[-6,9],[-5,11],[-4,13],[-3,15],[-2,17],[-1,19]],\"t\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0036","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23}，函數表為 {(-12, 23), (-11, 1), (-10, 3), (-9, 5), (-8, 7), (-7, 9), (-6, 11), (-5, 13), (-4, 15), (-3, 17), (-2, 19), (-1, 21)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(17)，其中 17∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(17)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054036,"parameters":{"A":[-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23],"pairs":[[-12,23],[-11,1],[-10,3],[-9,5],[-8,7],[-7,9],[-6,11],[-5,13],[-4,15],[-3,17],[-2,19],[-1,21]],"t":17,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23],\"operation\":\"inverse_evaluate\",\"pairs\":[[-12,23],[-11,1],[-10,3],[-9,5],[-8,7],[-7,9],[-6,11],[-5,13],[-4,15],[-3,17],[-2,19],[-1,21]],\"t\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0037","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}，函數表為 {(-13, 25), (-12, 1), (-11, 3), (-10, 5), (-9, 7), (-8, 9), (-7, 11), (-6, 13), (-5, 15), (-4, 17), (-3, 19), (-2, 21), (-1, 23)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(19)，其中 19∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(19)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054037,"parameters":{"A":[-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25],"pairs":[[-13,25],[-12,1],[-11,3],[-10,5],[-9,7],[-8,9],[-7,11],[-6,13],[-5,15],[-4,17],[-3,19],[-2,21],[-1,23]],"t":19,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25],\"operation\":\"inverse_evaluate\",\"pairs\":[[-13,25],[-12,1],[-11,3],[-10,5],[-9,7],[-8,9],[-7,11],[-6,13],[-5,15],[-4,17],[-3,19],[-2,21],[-1,23]],\"t\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0038","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27}，函數表為 {(-14, 27), (-13, 1), (-12, 3), (-11, 5), (-10, 7), (-9, 9), (-8, 11), (-7, 13), (-6, 15), (-5, 17), (-4, 19), (-3, 21), (-2, 23), (-1, 25)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(21)，其中 21∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(21)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054038,"parameters":{"A":[-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27],"pairs":[[-14,27],[-13,1],[-12,3],[-11,5],[-10,7],[-9,9],[-8,11],[-7,13],[-6,15],[-5,17],[-4,19],[-3,21],[-2,23],[-1,25]],"t":21,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27],\"operation\":\"inverse_evaluate\",\"pairs\":[[-14,27],[-13,1],[-12,3],[-11,5],[-10,7],[-9,9],[-8,11],[-7,13],[-6,15],[-5,17],[-4,19],[-3,21],[-2,23],[-1,25]],\"t\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0039","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}，函數表為 {(-15, 29), (-14, 1), (-13, 3), (-12, 5), (-11, 7), (-10, 9), (-9, 11), (-8, 13), (-7, 15), (-6, 17), (-5, 19), (-4, 21), (-3, 23), (-2, 25), (-1, 27)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(23)，其中 23∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(23)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054039,"parameters":{"A":[-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29],"pairs":[[-15,29],[-14,1],[-13,3],[-12,5],[-11,7],[-10,9],[-9,11],[-8,13],[-7,15],[-6,17],[-5,19],[-4,21],[-3,23],[-2,25],[-1,27]],"t":23,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29],\"operation\":\"inverse_evaluate\",\"pairs\":[[-15,29],[-14,1],[-13,3],[-12,5],[-11,7],[-10,9],[-9,11],[-8,13],[-7,15],[-6,17],[-5,19],[-4,21],[-3,23],[-2,25],[-1,27]],\"t\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0040","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31}，函數表為 {(-16, 31), (-15, 1), (-14, 3), (-13, 5), (-12, 7), (-11, 9), (-10, 11), (-9, 13), (-8, 15), (-7, 17), (-6, 19), (-5, 21), (-4, 23), (-3, 25), (-2, 27), (-1, 29)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(25)，其中 25∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(25)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054040,"parameters":{"A":[-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31],"pairs":[[-16,31],[-15,1],[-14,3],[-13,5],[-12,7],[-11,9],[-10,11],[-9,13],[-8,15],[-7,17],[-6,19],[-5,21],[-4,23],[-3,25],[-2,27],[-1,29]],"t":25,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31],\"operation\":\"inverse_evaluate\",\"pairs\":[[-16,31],[-15,1],[-14,3],[-13,5],[-12,7],[-11,9],[-10,11],[-9,13],[-8,15],[-7,17],[-6,19],[-5,21],[-4,23],[-3,25],[-2,27],[-1,29]],\"t\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0041","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}，函數表為 {(-17, 33), (-16, 1), (-15, 3), (-14, 5), (-13, 7), (-12, 9), (-11, 11), (-10, 13), (-9, 15), (-8, 17), (-7, 19), (-6, 21), (-5, 23), (-4, 25), (-3, 27), (-2, 29), (-1, 31)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(27)，其中 27∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(27)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054041,"parameters":{"A":[-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33],"pairs":[[-17,33],[-16,1],[-15,3],[-14,5],[-13,7],[-12,9],[-11,11],[-10,13],[-9,15],[-8,17],[-7,19],[-6,21],[-5,23],[-4,25],[-3,27],[-2,29],[-1,31]],"t":27,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33],\"operation\":\"inverse_evaluate\",\"pairs\":[[-17,33],[-16,1],[-15,3],[-14,5],[-13,7],[-12,9],[-11,11],[-10,13],[-9,15],[-8,17],[-7,19],[-6,21],[-5,23],[-4,25],[-3,27],[-2,29],[-1,31]],\"t\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0042","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35}，函數表為 {(-18, 35), (-17, 1), (-16, 3), (-15, 5), (-14, 7), (-13, 9), (-12, 11), (-11, 13), (-10, 15), (-9, 17), (-8, 19), (-7, 21), (-6, 23), (-5, 25), (-4, 27), (-3, 29), (-2, 31), (-1, 33)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(29)，其中 29∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(29)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054042,"parameters":{"A":[-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35],"pairs":[[-18,35],[-17,1],[-16,3],[-15,5],[-14,7],[-13,9],[-12,11],[-11,13],[-10,15],[-9,17],[-8,19],[-7,21],[-6,23],[-5,25],[-4,27],[-3,29],[-2,31],[-1,33]],"t":29,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35],\"operation\":\"inverse_evaluate\",\"pairs\":[[-18,35],[-17,1],[-16,3],[-15,5],[-14,7],[-13,9],[-12,11],[-11,13],[-10,15],[-9,17],[-8,19],[-7,21],[-6,23],[-5,25],[-4,27],[-3,29],[-2,31],[-1,33]],\"t\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0043","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37}，函數表為 {(-19, 37), (-18, 1), (-17, 3), (-16, 5), (-15, 7), (-14, 9), (-13, 11), (-12, 13), (-11, 15), (-10, 17), (-9, 19), (-8, 21), (-7, 23), (-6, 25), (-5, 27), (-4, 29), (-3, 31), (-2, 33), (-1, 35)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(31)，其中 31∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(31)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054043,"parameters":{"A":[-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37],"pairs":[[-19,37],[-18,1],[-17,3],[-16,5],[-15,7],[-14,9],[-13,11],[-12,13],[-11,15],[-10,17],[-9,19],[-8,21],[-7,23],[-6,25],[-5,27],[-4,29],[-3,31],[-2,33],[-1,35]],"t":31,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37],\"operation\":\"inverse_evaluate\",\"pairs\":[[-19,37],[-18,1],[-17,3],[-16,5],[-15,7],[-14,9],[-13,11],[-12,13],[-11,15],[-10,17],[-9,19],[-8,21],[-7,23],[-6,25],[-5,27],[-4,29],[-3,31],[-2,33],[-1,35]],\"t\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0044","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39}，函數表為 {(-20, 39), (-19, 1), (-18, 3), (-17, 5), (-16, 7), (-15, 9), (-14, 11), (-13, 13), (-12, 15), (-11, 17), (-10, 19), (-9, 21), (-8, 23), (-7, 25), (-6, 27), (-5, 29), (-4, 31), (-3, 33), (-2, 35), (-1, 37)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(33)，其中 33∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(33)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054044,"parameters":{"A":[-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39],"pairs":[[-20,39],[-19,1],[-18,3],[-17,5],[-16,7],[-15,9],[-14,11],[-13,13],[-12,15],[-11,17],[-10,19],[-9,21],[-8,23],[-7,25],[-6,27],[-5,29],[-4,31],[-3,33],[-2,35],[-1,37]],"t":33,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39],\"operation\":\"inverse_evaluate\",\"pairs\":[[-20,39],[-19,1],[-18,3],[-17,5],[-16,7],[-15,9],[-14,11],[-13,13],[-12,15],[-11,17],[-10,19],[-9,21],[-8,23],[-7,25],[-6,27],[-5,29],[-4,31],[-3,33],[-2,35],[-1,37]],\"t\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0045","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41}，函數表為 {(-21, 41), (-20, 1), (-19, 3), (-18, 5), (-17, 7), (-16, 9), (-15, 11), (-14, 13), (-13, 15), (-12, 17), (-11, 19), (-10, 21), (-9, 23), (-8, 25), (-7, 27), (-6, 29), (-5, 31), (-4, 33), (-3, 35), (-2, 37), (-1, 39)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(35)，其中 35∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(35)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054045,"parameters":{"A":[-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41],"pairs":[[-21,41],[-20,1],[-19,3],[-18,5],[-17,7],[-16,9],[-15,11],[-14,13],[-13,15],[-12,17],[-11,19],[-10,21],[-9,23],[-8,25],[-7,27],[-6,29],[-5,31],[-4,33],[-3,35],[-2,37],[-1,39]],"t":35,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41],\"operation\":\"inverse_evaluate\",\"pairs\":[[-21,41],[-20,1],[-19,3],[-18,5],[-17,7],[-16,9],[-15,11],[-14,13],[-13,15],[-12,17],[-11,19],[-10,21],[-9,23],[-8,25],[-7,27],[-6,29],[-5,31],[-4,33],[-3,35],[-2,37],[-1,39]],\"t\":35}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0046","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43}，函數表為 {(-22, 43), (-21, 1), (-20, 3), (-19, 5), (-18, 7), (-17, 9), (-16, 11), (-15, 13), (-14, 15), (-13, 17), (-12, 19), (-11, 21), (-10, 23), (-9, 25), (-8, 27), (-7, 29), (-6, 31), (-5, 33), (-4, 35), (-3, 37), (-2, 39), (-1, 41)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(37)，其中 37∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(37)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054046,"parameters":{"A":[-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43],"pairs":[[-22,43],[-21,1],[-20,3],[-19,5],[-18,7],[-17,9],[-16,11],[-15,13],[-14,15],[-13,17],[-12,19],[-11,21],[-10,23],[-9,25],[-8,27],[-7,29],[-6,31],[-5,33],[-4,35],[-3,37],[-2,39],[-1,41]],"t":37,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43],\"operation\":\"inverse_evaluate\",\"pairs\":[[-22,43],[-21,1],[-20,3],[-19,5],[-18,7],[-17,9],[-16,11],[-15,13],[-14,15],[-13,17],[-12,19],[-11,21],[-10,23],[-9,25],[-8,27],[-7,29],[-6,31],[-5,33],[-4,35],[-3,37],[-2,39],[-1,41]],\"t\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0047","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45}，函數表為 {(-23, 45), (-22, 1), (-21, 3), (-20, 5), (-19, 7), (-18, 9), (-17, 11), (-16, 13), (-15, 15), (-14, 17), (-13, 19), (-12, 21), (-11, 23), (-10, 25), (-9, 27), (-8, 29), (-7, 31), (-6, 33), (-5, 35), (-4, 37), (-3, 39), (-2, 41), (-1, 43)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(39)，其中 39∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(39)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054047,"parameters":{"A":[-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45],"pairs":[[-23,45],[-22,1],[-21,3],[-20,5],[-19,7],[-18,9],[-17,11],[-16,13],[-15,15],[-14,17],[-13,19],[-12,21],[-11,23],[-10,25],[-9,27],[-8,29],[-7,31],[-6,33],[-5,35],[-4,37],[-3,39],[-2,41],[-1,43]],"t":39,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45],\"operation\":\"inverse_evaluate\",\"pairs\":[[-23,45],[-22,1],[-21,3],[-20,5],[-19,7],[-18,9],[-17,11],[-16,13],[-15,15],[-14,17],[-13,19],[-12,21],[-11,23],[-10,25],[-9,27],[-8,29],[-7,31],[-6,33],[-5,35],[-4,37],[-3,39],[-2,41],[-1,43]],\"t\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0048","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47}，函數表為 {(-24, 47), (-23, 1), (-22, 3), (-21, 5), (-20, 7), (-19, 9), (-18, 11), (-17, 13), (-16, 15), (-15, 17), (-14, 19), (-13, 21), (-12, 23), (-11, 25), (-10, 27), (-9, 29), (-8, 31), (-7, 33), (-6, 35), (-5, 37), (-4, 39), (-3, 41), (-2, 43), (-1, 45)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(41)，其中 41∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(41)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054048,"parameters":{"A":[-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47],"pairs":[[-24,47],[-23,1],[-22,3],[-21,5],[-20,7],[-19,9],[-18,11],[-17,13],[-16,15],[-15,17],[-14,19],[-13,21],[-12,23],[-11,25],[-10,27],[-9,29],[-8,31],[-7,33],[-6,35],[-5,37],[-4,39],[-3,41],[-2,43],[-1,45]],"t":41,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47],\"operation\":\"inverse_evaluate\",\"pairs\":[[-24,47],[-23,1],[-22,3],[-21,5],[-20,7],[-19,9],[-18,11],[-17,13],[-16,15],[-15,17],[-14,19],[-13,21],[-12,23],[-11,25],[-10,27],[-9,29],[-8,31],[-7,33],[-6,35],[-5,37],[-4,39],[-3,41],[-2,43],[-1,45]],\"t\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0049","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49}，函數表為 {(-25, 49), (-24, 1), (-23, 3), (-22, 5), (-21, 7), (-20, 9), (-19, 11), (-18, 13), (-17, 15), (-16, 17), (-15, 19), (-14, 21), (-13, 23), (-12, 25), (-11, 27), (-10, 29), (-9, 31), (-8, 33), (-7, 35), (-6, 37), (-5, 39), (-4, 41), (-3, 43), (-2, 45), (-1, 47)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(43)，其中 43∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(43)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054049,"parameters":{"A":[-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49],"pairs":[[-25,49],[-24,1],[-23,3],[-22,5],[-21,7],[-20,9],[-19,11],[-18,13],[-17,15],[-16,17],[-15,19],[-14,21],[-13,23],[-12,25],[-11,27],[-10,29],[-9,31],[-8,33],[-7,35],[-6,37],[-5,39],[-4,41],[-3,43],[-2,45],[-1,47]],"t":43,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49],\"operation\":\"inverse_evaluate\",\"pairs\":[[-25,49],[-24,1],[-23,3],[-22,5],[-21,7],[-20,9],[-19,11],[-18,13],[-17,15],[-16,17],[-15,19],[-14,21],[-13,23],[-12,25],[-11,27],[-10,29],[-9,31],[-8,33],[-7,35],[-6,37],[-5,39],[-4,41],[-3,43],[-2,45],[-1,47]],\"t\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0050","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"反函數在合法目標的值","stem_zh":"明示總函數 f:A→B，A={-26, -25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51}，函數表為 {(-26, 51), (-25, 1), (-24, 3), (-23, 5), (-22, 7), (-21, 9), (-20, 11), (-19, 13), (-18, 15), (-17, 17), (-16, 19), (-15, 21), (-14, 23), (-13, 25), (-12, 27), (-11, 29), (-10, 31), (-9, 33), (-8, 35), (-7, 37), (-6, 39), (-5, 41), (-4, 43), (-3, 45), (-2, 47), (-1, 49)}，且 f 已確認為雙射。令 g=f⁻¹，求 g(45)，其中 45∈B。","answer":{"kind":"short_text","value":"-3","explanation_zh":"g(45)=-3。反函數值是 f 映到此目標的唯一合法輸入，不是目標的倒數或 1/f(x)。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["反函數在合法目標的值"]},"variant":{"family_id":"qf-inverse-function-evaluation","seed":202610054050,"parameters":{"A":[-26,-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51],"pairs":[[-26,51],[-25,1],[-24,3],[-23,5],[-22,7],[-21,9],[-20,11],[-19,13],[-18,15],[-17,17],[-16,19],[-15,21],[-14,23],[-13,25],[-12,27],[-11,29],[-10,31],[-9,33],[-8,35],[-7,37],[-6,39],[-5,41],[-4,43],[-3,45],[-2,47],[-1,49]],"t":45,"operation":"inverse_evaluate","semantic_template":"qf-inverse-function-evaluation/{\"A\":[-26,-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51],\"operation\":\"inverse_evaluate\",\"pairs\":[[-26,51],[-25,1],[-24,3],[-23,5],[-22,7],[-21,9],[-20,11],[-19,13],[-18,15],[-17,17],[-16,19],[-15,21],[-14,23],[-13,25],[-12,27],[-11,29],[-10,31],[-9,33],[-8,35],[-7,37],[-6,39],[-5,41],[-4,43],[-3,45],[-2,47],[-1,49]],\"t\":45}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0051","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A=∅、B=∅，函數表為 ∅，且 f 已確認為雙射。候選總函數 g:B→A 的表為 ∅；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054051,"parameters":{"A":[],"B":[],"pairs":[],"G":[],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[],\"B\":[],\"G\":[],\"operation\":\"inverse_candidate\",\"pairs\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0052","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-3, -2, -1}、B={1, 3, 5}，函數表為 {(-3, 5), (-2, 1), (-1, 3)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -1), (3, -3), (5, -2)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054052,"parameters":{"A":[-3,-2,-1],"B":[1,3,5],"pairs":[[-3,5],[-2,1],[-1,3]],"G":[[1,-1],[3,-3],[5,-2]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-3,-2,-1],\"B\":[1,3,5],\"G\":[[1,-1],[3,-3],[5,-2]],\"operation\":\"inverse_candidate\",\"pairs\":[[-3,5],[-2,1],[-1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0053","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-4, -3, -2, -1}、B={1, 3, 5, 7}，函數表為 {(-4, 7), (-3, 1), (-2, 3), (-1, 5)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -3), (3, -2), (5, -1), (7, -4)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054053,"parameters":{"A":[-4,-3,-2,-1],"B":[1,3,5,7],"pairs":[[-4,7],[-3,1],[-2,3],[-1,5]],"G":[[1,-3],[3,-2],[5,-1],[7,-4]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-4,-3,-2,-1],\"B\":[1,3,5,7],\"G\":[[1,-3],[3,-2],[5,-1],[7,-4]],\"operation\":\"inverse_candidate\",\"pairs\":[[-4,7],[-3,1],[-2,3],[-1,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0054","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9}，函數表為 {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -3), (3, -2), (5, -1), (7, -5), (9, -4)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054054,"parameters":{"A":[-5,-4,-3,-2,-1],"B":[1,3,5,7,9],"pairs":[[-5,9],[-4,1],[-3,3],[-2,5],[-1,7]],"G":[[1,-3],[3,-2],[5,-1],[7,-5],[9,-4]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9],\"G\":[[1,-3],[3,-2],[5,-1],[7,-5],[9,-4]],\"operation\":\"inverse_candidate\",\"pairs\":[[-5,9],[-4,1],[-3,3],[-2,5],[-1,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0055","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11}，函數表為 {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -6)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054055,"parameters":{"A":[-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11],"pairs":[[-6,11],[-5,1],[-4,3],[-3,5],[-2,7],[-1,9]],"G":[[1,-5],[3,-4],[5,-3],[7,-2],[9,-1],[11,-6]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11],\"G\":[[1,-5],[3,-4],[5,-3],[7,-2],[9,-1],[11,-6]],\"operation\":\"inverse_candidate\",\"pairs\":[[-6,11],[-5,1],[-4,3],[-3,5],[-2,7],[-1,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0056","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13}，函數表為 {(-7, 13), (-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -7), (13, -6)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054056,"parameters":{"A":[-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13],"pairs":[[-7,13],[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11]],"G":[[1,-5],[3,-4],[5,-3],[7,-2],[9,-1],[11,-7],[13,-6]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13],\"G\":[[1,-5],[3,-4],[5,-3],[7,-2],[9,-1],[11,-7],[13,-6]],\"operation\":\"inverse_candidate\",\"pairs\":[[-7,13],[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0057","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15}，函數表為 {(-8, 15), (-7, 1), (-6, 3), (-5, 5), (-4, 7), (-3, 9), (-2, 11), (-1, 13)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -8)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054057,"parameters":{"A":[-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15],"pairs":[[-8,15],[-7,1],[-6,3],[-5,5],[-4,7],[-3,9],[-2,11],[-1,13]],"G":[[1,-7],[3,-6],[5,-5],[7,-4],[9,-3],[11,-2],[13,-1],[15,-8]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15],\"G\":[[1,-7],[3,-6],[5,-5],[7,-4],[9,-3],[11,-2],[13,-1],[15,-8]],\"operation\":\"inverse_candidate\",\"pairs\":[[-8,15],[-7,1],[-6,3],[-5,5],[-4,7],[-3,9],[-2,11],[-1,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0058","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17}，函數表為 {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -9), (17, -8)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054058,"parameters":{"A":[-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17],"pairs":[[-9,17],[-8,1],[-7,3],[-6,5],[-5,7],[-4,9],[-3,11],[-2,13],[-1,15]],"G":[[1,-7],[3,-6],[5,-5],[7,-4],[9,-3],[11,-2],[13,-1],[15,-9],[17,-8]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17],\"G\":[[1,-7],[3,-6],[5,-5],[7,-4],[9,-3],[11,-2],[13,-1],[15,-9],[17,-8]],\"operation\":\"inverse_candidate\",\"pairs\":[[-9,17],[-8,1],[-7,3],[-6,5],[-5,7],[-4,9],[-3,11],[-2,13],[-1,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0059","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}，函數表為 {(-10, 19), (-9, 1), (-8, 3), (-7, 5), (-6, 7), (-5, 9), (-4, 11), (-3, 13), (-2, 15), (-1, 17)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -10)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054059,"parameters":{"A":[-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19],"pairs":[[-10,19],[-9,1],[-8,3],[-7,5],[-6,7],[-5,9],[-4,11],[-3,13],[-2,15],[-1,17]],"G":[[1,-9],[3,-8],[5,-7],[7,-6],[9,-5],[11,-4],[13,-3],[15,-2],[17,-1],[19,-10]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19],\"G\":[[1,-9],[3,-8],[5,-7],[7,-6],[9,-5],[11,-4],[13,-3],[15,-2],[17,-1],[19,-10]],\"operation\":\"inverse_candidate\",\"pairs\":[[-10,19],[-9,1],[-8,3],[-7,5],[-6,7],[-5,9],[-4,11],[-3,13],[-2,15],[-1,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0060","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}，函數表為 {(-11, 21), (-10, 1), (-9, 3), (-8, 5), (-7, 7), (-6, 9), (-5, 11), (-4, 13), (-3, 15), (-2, 17), (-1, 19)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -11), (21, -10)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054060,"parameters":{"A":[-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21],"pairs":[[-11,21],[-10,1],[-9,3],[-8,5],[-7,7],[-6,9],[-5,11],[-4,13],[-3,15],[-2,17],[-1,19]],"G":[[1,-9],[3,-8],[5,-7],[7,-6],[9,-5],[11,-4],[13,-3],[15,-2],[17,-1],[19,-11],[21,-10]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21],\"G\":[[1,-9],[3,-8],[5,-7],[7,-6],[9,-5],[11,-4],[13,-3],[15,-2],[17,-1],[19,-11],[21,-10]],\"operation\":\"inverse_candidate\",\"pairs\":[[-11,21],[-10,1],[-9,3],[-8,5],[-7,7],[-6,9],[-5,11],[-4,13],[-3,15],[-2,17],[-1,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0061","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23}，函數表為 {(-12, 23), (-11, 1), (-10, 3), (-9, 5), (-8, 7), (-7, 9), (-6, 11), (-5, 13), (-4, 15), (-3, 17), (-2, 19), (-1, 21)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -12)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054061,"parameters":{"A":[-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23],"pairs":[[-12,23],[-11,1],[-10,3],[-9,5],[-8,7],[-7,9],[-6,11],[-5,13],[-4,15],[-3,17],[-2,19],[-1,21]],"G":[[1,-11],[3,-10],[5,-9],[7,-8],[9,-7],[11,-6],[13,-5],[15,-4],[17,-3],[19,-2],[21,-1],[23,-12]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23],\"G\":[[1,-11],[3,-10],[5,-9],[7,-8],[9,-7],[11,-6],[13,-5],[15,-4],[17,-3],[19,-2],[21,-1],[23,-12]],\"operation\":\"inverse_candidate\",\"pairs\":[[-12,23],[-11,1],[-10,3],[-9,5],[-8,7],[-7,9],[-6,11],[-5,13],[-4,15],[-3,17],[-2,19],[-1,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0062","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}，函數表為 {(-13, 25), (-12, 1), (-11, 3), (-10, 5), (-9, 7), (-8, 9), (-7, 11), (-6, 13), (-5, 15), (-4, 17), (-3, 19), (-2, 21), (-1, 23)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -13), (25, -12)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054062,"parameters":{"A":[-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25],"pairs":[[-13,25],[-12,1],[-11,3],[-10,5],[-9,7],[-8,9],[-7,11],[-6,13],[-5,15],[-4,17],[-3,19],[-2,21],[-1,23]],"G":[[1,-11],[3,-10],[5,-9],[7,-8],[9,-7],[11,-6],[13,-5],[15,-4],[17,-3],[19,-2],[21,-1],[23,-13],[25,-12]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25],\"G\":[[1,-11],[3,-10],[5,-9],[7,-8],[9,-7],[11,-6],[13,-5],[15,-4],[17,-3],[19,-2],[21,-1],[23,-13],[25,-12]],\"operation\":\"inverse_candidate\",\"pairs\":[[-13,25],[-12,1],[-11,3],[-10,5],[-9,7],[-8,9],[-7,11],[-6,13],[-5,15],[-4,17],[-3,19],[-2,21],[-1,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0063","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27}，函數表為 {(-14, 27), (-13, 1), (-12, 3), (-11, 5), (-10, 7), (-9, 9), (-8, 11), (-7, 13), (-6, 15), (-5, 17), (-4, 19), (-3, 21), (-2, 23), (-1, 25)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -13), (3, -12), (5, -11), (7, -10), (9, -9), (11, -8), (13, -7), (15, -6), (17, -5), (19, -4), (21, -3), (23, -2), (25, -1), (27, -14)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054063,"parameters":{"A":[-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27],"pairs":[[-14,27],[-13,1],[-12,3],[-11,5],[-10,7],[-9,9],[-8,11],[-7,13],[-6,15],[-5,17],[-4,19],[-3,21],[-2,23],[-1,25]],"G":[[1,-13],[3,-12],[5,-11],[7,-10],[9,-9],[11,-8],[13,-7],[15,-6],[17,-5],[19,-4],[21,-3],[23,-2],[25,-1],[27,-14]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27],\"G\":[[1,-13],[3,-12],[5,-11],[7,-10],[9,-9],[11,-8],[13,-7],[15,-6],[17,-5],[19,-4],[21,-3],[23,-2],[25,-1],[27,-14]],\"operation\":\"inverse_candidate\",\"pairs\":[[-14,27],[-13,1],[-12,3],[-11,5],[-10,7],[-9,9],[-8,11],[-7,13],[-6,15],[-5,17],[-4,19],[-3,21],[-2,23],[-1,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0064","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}，函數表為 {(-15, 29), (-14, 1), (-13, 3), (-12, 5), (-11, 7), (-10, 9), (-9, 11), (-8, 13), (-7, 15), (-6, 17), (-5, 19), (-4, 21), (-3, 23), (-2, 25), (-1, 27)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -13), (3, -12), (5, -11), (7, -10), (9, -9), (11, -8), (13, -7), (15, -6), (17, -5), (19, -4), (21, -3), (23, -2), (25, -1), (27, -15), (29, -14)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054064,"parameters":{"A":[-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29],"pairs":[[-15,29],[-14,1],[-13,3],[-12,5],[-11,7],[-10,9],[-9,11],[-8,13],[-7,15],[-6,17],[-5,19],[-4,21],[-3,23],[-2,25],[-1,27]],"G":[[1,-13],[3,-12],[5,-11],[7,-10],[9,-9],[11,-8],[13,-7],[15,-6],[17,-5],[19,-4],[21,-3],[23,-2],[25,-1],[27,-15],[29,-14]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29],\"G\":[[1,-13],[3,-12],[5,-11],[7,-10],[9,-9],[11,-8],[13,-7],[15,-6],[17,-5],[19,-4],[21,-3],[23,-2],[25,-1],[27,-15],[29,-14]],\"operation\":\"inverse_candidate\",\"pairs\":[[-15,29],[-14,1],[-13,3],[-12,5],[-11,7],[-10,9],[-9,11],[-8,13],[-7,15],[-6,17],[-5,19],[-4,21],[-3,23],[-2,25],[-1,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0065","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31}，函數表為 {(-16, 31), (-15, 1), (-14, 3), (-13, 5), (-12, 7), (-11, 9), (-10, 11), (-9, 13), (-8, 15), (-7, 17), (-6, 19), (-5, 21), (-4, 23), (-3, 25), (-2, 27), (-1, 29)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -16)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054065,"parameters":{"A":[-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31],"pairs":[[-16,31],[-15,1],[-14,3],[-13,5],[-12,7],[-11,9],[-10,11],[-9,13],[-8,15],[-7,17],[-6,19],[-5,21],[-4,23],[-3,25],[-2,27],[-1,29]],"G":[[1,-15],[3,-14],[5,-13],[7,-12],[9,-11],[11,-10],[13,-9],[15,-8],[17,-7],[19,-6],[21,-5],[23,-4],[25,-3],[27,-2],[29,-1],[31,-16]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31],\"G\":[[1,-15],[3,-14],[5,-13],[7,-12],[9,-11],[11,-10],[13,-9],[15,-8],[17,-7],[19,-6],[21,-5],[23,-4],[25,-3],[27,-2],[29,-1],[31,-16]],\"operation\":\"inverse_candidate\",\"pairs\":[[-16,31],[-15,1],[-14,3],[-13,5],[-12,7],[-11,9],[-10,11],[-9,13],[-8,15],[-7,17],[-6,19],[-5,21],[-4,23],[-3,25],[-2,27],[-1,29]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0066","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}，函數表為 {(-17, 33), (-16, 1), (-15, 3), (-14, 5), (-13, 7), (-12, 9), (-11, 11), (-10, 13), (-9, 15), (-8, 17), (-7, 19), (-6, 21), (-5, 23), (-4, 25), (-3, 27), (-2, 29), (-1, 31)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -17), (33, -16)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054066,"parameters":{"A":[-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33],"pairs":[[-17,33],[-16,1],[-15,3],[-14,5],[-13,7],[-12,9],[-11,11],[-10,13],[-9,15],[-8,17],[-7,19],[-6,21],[-5,23],[-4,25],[-3,27],[-2,29],[-1,31]],"G":[[1,-15],[3,-14],[5,-13],[7,-12],[9,-11],[11,-10],[13,-9],[15,-8],[17,-7],[19,-6],[21,-5],[23,-4],[25,-3],[27,-2],[29,-1],[31,-17],[33,-16]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33],\"G\":[[1,-15],[3,-14],[5,-13],[7,-12],[9,-11],[11,-10],[13,-9],[15,-8],[17,-7],[19,-6],[21,-5],[23,-4],[25,-3],[27,-2],[29,-1],[31,-17],[33,-16]],\"operation\":\"inverse_candidate\",\"pairs\":[[-17,33],[-16,1],[-15,3],[-14,5],[-13,7],[-12,9],[-11,11],[-10,13],[-9,15],[-8,17],[-7,19],[-6,21],[-5,23],[-4,25],[-3,27],[-2,29],[-1,31]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0067","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35}，函數表為 {(-18, 35), (-17, 1), (-16, 3), (-15, 5), (-14, 7), (-13, 9), (-12, 11), (-11, 13), (-10, 15), (-9, 17), (-8, 19), (-7, 21), (-6, 23), (-5, 25), (-4, 27), (-3, 29), (-2, 31), (-1, 33)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -17), (3, -16), (5, -15), (7, -14), (9, -13), (11, -12), (13, -11), (15, -10), (17, -9), (19, -8), (21, -7), (23, -6), (25, -5), (27, -4), (29, -3), (31, -2), (33, -1), (35, -18)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054067,"parameters":{"A":[-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35],"pairs":[[-18,35],[-17,1],[-16,3],[-15,5],[-14,7],[-13,9],[-12,11],[-11,13],[-10,15],[-9,17],[-8,19],[-7,21],[-6,23],[-5,25],[-4,27],[-3,29],[-2,31],[-1,33]],"G":[[1,-17],[3,-16],[5,-15],[7,-14],[9,-13],[11,-12],[13,-11],[15,-10],[17,-9],[19,-8],[21,-7],[23,-6],[25,-5],[27,-4],[29,-3],[31,-2],[33,-1],[35,-18]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35],\"G\":[[1,-17],[3,-16],[5,-15],[7,-14],[9,-13],[11,-12],[13,-11],[15,-10],[17,-9],[19,-8],[21,-7],[23,-6],[25,-5],[27,-4],[29,-3],[31,-2],[33,-1],[35,-18]],\"operation\":\"inverse_candidate\",\"pairs\":[[-18,35],[-17,1],[-16,3],[-15,5],[-14,7],[-13,9],[-12,11],[-11,13],[-10,15],[-9,17],[-8,19],[-7,21],[-6,23],[-5,25],[-4,27],[-3,29],[-2,31],[-1,33]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0068","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37}，函數表為 {(-19, 37), (-18, 1), (-17, 3), (-16, 5), (-15, 7), (-14, 9), (-13, 11), (-12, 13), (-11, 15), (-10, 17), (-9, 19), (-8, 21), (-7, 23), (-6, 25), (-5, 27), (-4, 29), (-3, 31), (-2, 33), (-1, 35)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -17), (3, -16), (5, -15), (7, -14), (9, -13), (11, -12), (13, -11), (15, -10), (17, -9), (19, -8), (21, -7), (23, -6), (25, -5), (27, -4), (29, -3), (31, -2), (33, -1), (35, -19), (37, -18)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054068,"parameters":{"A":[-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37],"pairs":[[-19,37],[-18,1],[-17,3],[-16,5],[-15,7],[-14,9],[-13,11],[-12,13],[-11,15],[-10,17],[-9,19],[-8,21],[-7,23],[-6,25],[-5,27],[-4,29],[-3,31],[-2,33],[-1,35]],"G":[[1,-17],[3,-16],[5,-15],[7,-14],[9,-13],[11,-12],[13,-11],[15,-10],[17,-9],[19,-8],[21,-7],[23,-6],[25,-5],[27,-4],[29,-3],[31,-2],[33,-1],[35,-19],[37,-18]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37],\"G\":[[1,-17],[3,-16],[5,-15],[7,-14],[9,-13],[11,-12],[13,-11],[15,-10],[17,-9],[19,-8],[21,-7],[23,-6],[25,-5],[27,-4],[29,-3],[31,-2],[33,-1],[35,-19],[37,-18]],\"operation\":\"inverse_candidate\",\"pairs\":[[-19,37],[-18,1],[-17,3],[-16,5],[-15,7],[-14,9],[-13,11],[-12,13],[-11,15],[-10,17],[-9,19],[-8,21],[-7,23],[-6,25],[-5,27],[-4,29],[-3,31],[-2,33],[-1,35]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0069","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39}，函數表為 {(-20, 39), (-19, 1), (-18, 3), (-17, 5), (-16, 7), (-15, 9), (-14, 11), (-13, 13), (-12, 15), (-11, 17), (-10, 19), (-9, 21), (-8, 23), (-7, 25), (-6, 27), (-5, 29), (-4, 31), (-3, 33), (-2, 35), (-1, 37)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -19), (3, -18), (5, -17), (7, -16), (9, -15), (11, -14), (13, -13), (15, -12), (17, -11), (19, -10), (21, -9), (23, -8), (25, -7), (27, -6), (29, -5), (31, -4), (33, -3), (35, -2), (37, -1), (39, -20)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054069,"parameters":{"A":[-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39],"pairs":[[-20,39],[-19,1],[-18,3],[-17,5],[-16,7],[-15,9],[-14,11],[-13,13],[-12,15],[-11,17],[-10,19],[-9,21],[-8,23],[-7,25],[-6,27],[-5,29],[-4,31],[-3,33],[-2,35],[-1,37]],"G":[[1,-19],[3,-18],[5,-17],[7,-16],[9,-15],[11,-14],[13,-13],[15,-12],[17,-11],[19,-10],[21,-9],[23,-8],[25,-7],[27,-6],[29,-5],[31,-4],[33,-3],[35,-2],[37,-1],[39,-20]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39],\"G\":[[1,-19],[3,-18],[5,-17],[7,-16],[9,-15],[11,-14],[13,-13],[15,-12],[17,-11],[19,-10],[21,-9],[23,-8],[25,-7],[27,-6],[29,-5],[31,-4],[33,-3],[35,-2],[37,-1],[39,-20]],\"operation\":\"inverse_candidate\",\"pairs\":[[-20,39],[-19,1],[-18,3],[-17,5],[-16,7],[-15,9],[-14,11],[-13,13],[-12,15],[-11,17],[-10,19],[-9,21],[-8,23],[-7,25],[-6,27],[-5,29],[-4,31],[-3,33],[-2,35],[-1,37]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0070","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41}，函數表為 {(-21, 41), (-20, 1), (-19, 3), (-18, 5), (-17, 7), (-16, 9), (-15, 11), (-14, 13), (-13, 15), (-12, 17), (-11, 19), (-10, 21), (-9, 23), (-8, 25), (-7, 27), (-6, 29), (-5, 31), (-4, 33), (-3, 35), (-2, 37), (-1, 39)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -19), (3, -18), (5, -17), (7, -16), (9, -15), (11, -14), (13, -13), (15, -12), (17, -11), (19, -10), (21, -9), (23, -8), (25, -7), (27, -6), (29, -5), (31, -4), (33, -3), (35, -2), (37, -1), (39, -21), (41, -20)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054070,"parameters":{"A":[-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41],"pairs":[[-21,41],[-20,1],[-19,3],[-18,5],[-17,7],[-16,9],[-15,11],[-14,13],[-13,15],[-12,17],[-11,19],[-10,21],[-9,23],[-8,25],[-7,27],[-6,29],[-5,31],[-4,33],[-3,35],[-2,37],[-1,39]],"G":[[1,-19],[3,-18],[5,-17],[7,-16],[9,-15],[11,-14],[13,-13],[15,-12],[17,-11],[19,-10],[21,-9],[23,-8],[25,-7],[27,-6],[29,-5],[31,-4],[33,-3],[35,-2],[37,-1],[39,-21],[41,-20]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41],\"G\":[[1,-19],[3,-18],[5,-17],[7,-16],[9,-15],[11,-14],[13,-13],[15,-12],[17,-11],[19,-10],[21,-9],[23,-8],[25,-7],[27,-6],[29,-5],[31,-4],[33,-3],[35,-2],[37,-1],[39,-21],[41,-20]],\"operation\":\"inverse_candidate\",\"pairs\":[[-21,41],[-20,1],[-19,3],[-18,5],[-17,7],[-16,9],[-15,11],[-14,13],[-13,15],[-12,17],[-11,19],[-10,21],[-9,23],[-8,25],[-7,27],[-6,29],[-5,31],[-4,33],[-3,35],[-2,37],[-1,39]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0071","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43}，函數表為 {(-22, 43), (-21, 1), (-20, 3), (-19, 5), (-18, 7), (-17, 9), (-16, 11), (-15, 13), (-14, 15), (-13, 17), (-12, 19), (-11, 21), (-10, 23), (-9, 25), (-8, 27), (-7, 29), (-6, 31), (-5, 33), (-4, 35), (-3, 37), (-2, 39), (-1, 41)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -21), (3, -20), (5, -19), (7, -18), (9, -17), (11, -16), (13, -15), (15, -14), (17, -13), (19, -12), (21, -11), (23, -10), (25, -9), (27, -8), (29, -7), (31, -6), (33, -5), (35, -4), (37, -3), (39, -2), (41, -1), (43, -22)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054071,"parameters":{"A":[-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43],"pairs":[[-22,43],[-21,1],[-20,3],[-19,5],[-18,7],[-17,9],[-16,11],[-15,13],[-14,15],[-13,17],[-12,19],[-11,21],[-10,23],[-9,25],[-8,27],[-7,29],[-6,31],[-5,33],[-4,35],[-3,37],[-2,39],[-1,41]],"G":[[1,-21],[3,-20],[5,-19],[7,-18],[9,-17],[11,-16],[13,-15],[15,-14],[17,-13],[19,-12],[21,-11],[23,-10],[25,-9],[27,-8],[29,-7],[31,-6],[33,-5],[35,-4],[37,-3],[39,-2],[41,-1],[43,-22]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43],\"G\":[[1,-21],[3,-20],[5,-19],[7,-18],[9,-17],[11,-16],[13,-15],[15,-14],[17,-13],[19,-12],[21,-11],[23,-10],[25,-9],[27,-8],[29,-7],[31,-6],[33,-5],[35,-4],[37,-3],[39,-2],[41,-1],[43,-22]],\"operation\":\"inverse_candidate\",\"pairs\":[[-22,43],[-21,1],[-20,3],[-19,5],[-18,7],[-17,9],[-16,11],[-15,13],[-14,15],[-13,17],[-12,19],[-11,21],[-10,23],[-9,25],[-8,27],[-7,29],[-6,31],[-5,33],[-4,35],[-3,37],[-2,39],[-1,41]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0072","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45}，函數表為 {(-23, 45), (-22, 1), (-21, 3), (-20, 5), (-19, 7), (-18, 9), (-17, 11), (-16, 13), (-15, 15), (-14, 17), (-13, 19), (-12, 21), (-11, 23), (-10, 25), (-9, 27), (-8, 29), (-7, 31), (-6, 33), (-5, 35), (-4, 37), (-3, 39), (-2, 41), (-1, 43)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -21), (3, -20), (5, -19), (7, -18), (9, -17), (11, -16), (13, -15), (15, -14), (17, -13), (19, -12), (21, -11), (23, -10), (25, -9), (27, -8), (29, -7), (31, -6), (33, -5), (35, -4), (37, -3), (39, -2), (41, -1), (43, -23), (45, -22)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054072,"parameters":{"A":[-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45],"pairs":[[-23,45],[-22,1],[-21,3],[-20,5],[-19,7],[-18,9],[-17,11],[-16,13],[-15,15],[-14,17],[-13,19],[-12,21],[-11,23],[-10,25],[-9,27],[-8,29],[-7,31],[-6,33],[-5,35],[-4,37],[-3,39],[-2,41],[-1,43]],"G":[[1,-21],[3,-20],[5,-19],[7,-18],[9,-17],[11,-16],[13,-15],[15,-14],[17,-13],[19,-12],[21,-11],[23,-10],[25,-9],[27,-8],[29,-7],[31,-6],[33,-5],[35,-4],[37,-3],[39,-2],[41,-1],[43,-23],[45,-22]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45],\"G\":[[1,-21],[3,-20],[5,-19],[7,-18],[9,-17],[11,-16],[13,-15],[15,-14],[17,-13],[19,-12],[21,-11],[23,-10],[25,-9],[27,-8],[29,-7],[31,-6],[33,-5],[35,-4],[37,-3],[39,-2],[41,-1],[43,-23],[45,-22]],\"operation\":\"inverse_candidate\",\"pairs\":[[-23,45],[-22,1],[-21,3],[-20,5],[-19,7],[-18,9],[-17,11],[-16,13],[-15,15],[-14,17],[-13,19],[-12,21],[-11,23],[-10,25],[-9,27],[-8,29],[-7,31],[-6,33],[-5,35],[-4,37],[-3,39],[-2,41],[-1,43]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0073","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47}，函數表為 {(-24, 47), (-23, 1), (-22, 3), (-21, 5), (-20, 7), (-19, 9), (-18, 11), (-17, 13), (-16, 15), (-15, 17), (-14, 19), (-13, 21), (-12, 23), (-11, 25), (-10, 27), (-9, 29), (-8, 31), (-7, 33), (-6, 35), (-5, 37), (-4, 39), (-3, 41), (-2, 43), (-1, 45)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -23), (3, -22), (5, -21), (7, -20), (9, -19), (11, -18), (13, -17), (15, -16), (17, -15), (19, -14), (21, -13), (23, -12), (25, -11), (27, -10), (29, -9), (31, -8), (33, -7), (35, -6), (37, -5), (39, -4), (41, -3), (43, -2), (45, -1), (47, -24)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054073,"parameters":{"A":[-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47],"pairs":[[-24,47],[-23,1],[-22,3],[-21,5],[-20,7],[-19,9],[-18,11],[-17,13],[-16,15],[-15,17],[-14,19],[-13,21],[-12,23],[-11,25],[-10,27],[-9,29],[-8,31],[-7,33],[-6,35],[-5,37],[-4,39],[-3,41],[-2,43],[-1,45]],"G":[[1,-23],[3,-22],[5,-21],[7,-20],[9,-19],[11,-18],[13,-17],[15,-16],[17,-15],[19,-14],[21,-13],[23,-12],[25,-11],[27,-10],[29,-9],[31,-8],[33,-7],[35,-6],[37,-5],[39,-4],[41,-3],[43,-2],[45,-1],[47,-24]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47],\"G\":[[1,-23],[3,-22],[5,-21],[7,-20],[9,-19],[11,-18],[13,-17],[15,-16],[17,-15],[19,-14],[21,-13],[23,-12],[25,-11],[27,-10],[29,-9],[31,-8],[33,-7],[35,-6],[37,-5],[39,-4],[41,-3],[43,-2],[45,-1],[47,-24]],\"operation\":\"inverse_candidate\",\"pairs\":[[-24,47],[-23,1],[-22,3],[-21,5],[-20,7],[-19,9],[-18,11],[-17,13],[-16,15],[-15,17],[-14,19],[-13,21],[-12,23],[-11,25],[-10,27],[-9,29],[-8,31],[-7,33],[-6,35],[-5,37],[-4,39],[-3,41],[-2,43],[-1,45]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0074","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49}，函數表為 {(-25, 49), (-24, 1), (-23, 3), (-22, 5), (-21, 7), (-20, 9), (-19, 11), (-18, 13), (-17, 15), (-16, 17), (-15, 19), (-14, 21), (-13, 23), (-12, 25), (-11, 27), (-10, 29), (-9, 31), (-8, 33), (-7, 35), (-6, 37), (-5, 39), (-4, 41), (-3, 43), (-2, 45), (-1, 47)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -23), (3, -22), (5, -21), (7, -20), (9, -19), (11, -18), (13, -17), (15, -16), (17, -15), (19, -14), (21, -13), (23, -12), (25, -11), (27, -10), (29, -9), (31, -8), (33, -7), (35, -6), (37, -5), (39, -4), (41, -3), (43, -2), (45, -1), (47, -25), (49, -24)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"否","explanation_zh":"左還原=否；右還原=否。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054074,"parameters":{"A":[-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49],"pairs":[[-25,49],[-24,1],[-23,3],[-22,5],[-21,7],[-20,9],[-19,11],[-18,13],[-17,15],[-16,17],[-15,19],[-14,21],[-13,23],[-12,25],[-11,27],[-10,29],[-9,31],[-8,33],[-7,35],[-6,37],[-5,39],[-4,41],[-3,43],[-2,45],[-1,47]],"G":[[1,-23],[3,-22],[5,-21],[7,-20],[9,-19],[11,-18],[13,-17],[15,-16],[17,-15],[19,-14],[21,-13],[23,-12],[25,-11],[27,-10],[29,-9],[31,-8],[33,-7],[35,-6],[37,-5],[39,-4],[41,-3],[43,-2],[45,-1],[47,-25],[49,-24]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49],\"G\":[[1,-23],[3,-22],[5,-21],[7,-20],[9,-19],[11,-18],[13,-17],[15,-16],[17,-15],[19,-14],[21,-13],[23,-12],[25,-11],[27,-10],[29,-9],[31,-8],[33,-7],[35,-6],[37,-5],[39,-4],[41,-3],[43,-2],[45,-1],[47,-25],[49,-24]],\"operation\":\"inverse_candidate\",\"pairs\":[[-25,49],[-24,1],[-23,3],[-22,5],[-21,7],[-20,9],[-19,11],[-18,13],[-17,15],[-16,17],[-15,19],[-14,21],[-13,23],[-12,25],[-11,27],[-10,29],[-9,31],[-8,33],[-7,35],[-6,37],[-5,39],[-4,41],[-3,43],[-2,45],[-1,47]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0075","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"候選反函數的兩側還原","stem_zh":"明示總函數 f:A→B，A={-26, -25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}、B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51}，函數表為 {(-26, 51), (-25, 1), (-24, 3), (-23, 5), (-22, 7), (-21, 9), (-20, 11), (-19, 13), (-18, 15), (-17, 17), (-16, 19), (-15, 21), (-14, 23), (-13, 25), (-12, 27), (-11, 29), (-10, 31), (-9, 33), (-8, 35), (-7, 37), (-6, 39), (-5, 41), (-4, 43), (-3, 45), (-2, 47), (-1, 49)}，且 f 已確認為雙射。候選總函數 g:B→A 的表為 {(1, -25), (3, -24), (5, -23), (7, -22), (9, -21), (11, -20), (13, -19), (15, -18), (17, -17), (19, -16), (21, -15), (23, -14), (25, -13), (27, -12), (29, -11), (31, -10), (33, -9), (35, -8), (37, -7), (39, -6), (41, -5), (43, -4), (45, -3), (47, -2), (49, -1), (51, -26)}；逐側檢查兩個 identity，g 是否為 f 的反函數？","answer":{"kind":"short_text","value":"是","explanation_zh":"左還原=是；右還原=是。在明示型別下逐點確認 g(f(x))=x 與 f(g(y))=y；候選失敗不能靠少列輸入掩蓋。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選反函數的兩側還原"]},"variant":{"family_id":"qf-inverse-function-candidate","seed":202610054075,"parameters":{"A":[-26,-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],"B":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51],"pairs":[[-26,51],[-25,1],[-24,3],[-23,5],[-22,7],[-21,9],[-20,11],[-19,13],[-18,15],[-17,17],[-16,19],[-15,21],[-14,23],[-13,25],[-12,27],[-11,29],[-10,31],[-9,33],[-8,35],[-7,37],[-6,39],[-5,41],[-4,43],[-3,45],[-2,47],[-1,49]],"G":[[1,-25],[3,-24],[5,-23],[7,-22],[9,-21],[11,-20],[13,-19],[15,-18],[17,-17],[19,-16],[21,-15],[23,-14],[25,-13],[27,-12],[29,-11],[31,-10],[33,-9],[35,-8],[37,-7],[39,-6],[41,-5],[43,-4],[45,-3],[47,-2],[49,-1],[51,-26]],"operation":"inverse_candidate","semantic_template":"qf-inverse-function-candidate/{\"A\":[-26,-25,-24,-23,-22,-21,-20,-19,-18,-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1],\"B\":[1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51],\"G\":[[1,-25],[3,-24],[5,-23],[7,-22],[9,-21],[11,-20],[13,-19],[15,-18],[17,-17],[19,-16],[21,-15],[23,-14],[25,-13],[27,-12],[29,-11],[31,-10],[33,-9],[35,-8],[37,-7],[39,-6],[41,-5],[43,-4],[45,-3],[47,-2],[49,-1],[51,-26]],\"operation\":\"inverse_candidate\",\"pairs\":[[-26,51],[-25,1],[-24,3],[-23,5],[-22,7],[-21,9],[-20,11],[-19,13],[-18,15],[-17,17],[-16,19],[-15,21],[-14,23],[-13,25],[-12,27],[-11,29],[-10,31],[-9,33],[-8,35],[-7,37],[-6,39],[-5,41],[-4,43],[-3,45],[-2,47],[-1,49]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0076","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=1x+(-12)，m=1≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/1, 12/1)","explanation_zh":"g係數=(1/1, 12/1)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054076,"parameters":{"m":1,"b":-12,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-12,\"domain\":\"reals\",\"m\":1,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0077","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-2x+(-11)，m=-2≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/2, -11/2)","explanation_zh":"g係數=(-1/2, -11/2)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054077,"parameters":{"m":-2,"b":-11,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-11,\"domain\":\"reals\",\"m\":-2,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0078","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=3x+(-10)，m=3≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/3, 10/3)","explanation_zh":"g係數=(1/3, 10/3)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054078,"parameters":{"m":3,"b":-10,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-10,\"domain\":\"reals\",\"m\":3,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0079","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-4x+(-9)，m=-4≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/4, -9/4)","explanation_zh":"g係數=(-1/4, -9/4)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054079,"parameters":{"m":-4,"b":-9,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-9,\"domain\":\"reals\",\"m\":-4,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0080","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=5x+(-8)，m=5≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/5, 8/5)","explanation_zh":"g係數=(1/5, 8/5)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054080,"parameters":{"m":5,"b":-8,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-8,\"domain\":\"reals\",\"m\":5,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0081","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-6x+(-7)，m=-6≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/6, -7/6)","explanation_zh":"g係數=(-1/6, -7/6)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054081,"parameters":{"m":-6,"b":-7,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-7,\"domain\":\"reals\",\"m\":-6,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0082","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=7x+(-6)，m=7≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/7, 6/7)","explanation_zh":"g係數=(1/7, 6/7)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054082,"parameters":{"m":7,"b":-6,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-6,\"domain\":\"reals\",\"m\":7,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0083","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-8x+(-5)，m=-8≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/8, -5/8)","explanation_zh":"g係數=(-1/8, -5/8)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054083,"parameters":{"m":-8,"b":-5,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-5,\"domain\":\"reals\",\"m\":-8,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0084","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=9x+(-4)，m=9≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/9, 4/9)","explanation_zh":"g係數=(1/9, 4/9)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054084,"parameters":{"m":9,"b":-4,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-4,\"domain\":\"reals\",\"m\":9,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0085","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-10x+(-3)，m=-10≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/10, -3/10)","explanation_zh":"g係數=(-1/10, -3/10)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054085,"parameters":{"m":-10,"b":-3,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-3,\"domain\":\"reals\",\"m\":-10,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0086","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=11x+(-2)，m=11≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/11, 2/11)","explanation_zh":"g係數=(1/11, 2/11)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054086,"parameters":{"m":11,"b":-2,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-2,\"domain\":\"reals\",\"m\":11,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0087","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-12x+(-1)，m=-12≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/12, -1/12)","explanation_zh":"g係數=(-1/12, -1/12)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054087,"parameters":{"m":-12,"b":-1,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":-1,\"domain\":\"reals\",\"m\":-12,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0088","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=13x+(0)，m=13≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/13, 0/1)","explanation_zh":"g係數=(1/13, 0/1)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054088,"parameters":{"m":13,"b":0,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":0,\"domain\":\"reals\",\"m\":13,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0089","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-14x+(1)，m=-14≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/14, 1/14)","explanation_zh":"g係數=(-1/14, 1/14)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054089,"parameters":{"m":-14,"b":1,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":1,\"domain\":\"reals\",\"m\":-14,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0090","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=15x+(2)，m=15≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/15, -2/15)","explanation_zh":"g係數=(1/15, -2/15)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054090,"parameters":{"m":15,"b":2,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":2,\"domain\":\"reals\",\"m\":15,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0091","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-16x+(3)，m=-16≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/16, 3/16)","explanation_zh":"g係數=(-1/16, 3/16)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054091,"parameters":{"m":-16,"b":3,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":3,\"domain\":\"reals\",\"m\":-16,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0092","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=17x+(4)，m=17≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/17, -4/17)","explanation_zh":"g係數=(1/17, -4/17)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054092,"parameters":{"m":17,"b":4,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":4,\"domain\":\"reals\",\"m\":17,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0093","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-18x+(5)，m=-18≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/18, 5/18)","explanation_zh":"g係數=(-1/18, 5/18)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054093,"parameters":{"m":-18,"b":5,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":5,\"domain\":\"reals\",\"m\":-18,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0094","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=19x+(6)，m=19≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/19, -6/19)","explanation_zh":"g係數=(1/19, -6/19)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054094,"parameters":{"m":19,"b":6,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":6,\"domain\":\"reals\",\"m\":19,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0095","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-20x+(7)，m=-20≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/20, 7/20)","explanation_zh":"g係數=(-1/20, 7/20)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054095,"parameters":{"m":-20,"b":7,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":7,\"domain\":\"reals\",\"m\":-20,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0096","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=21x+(8)，m=21≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/21, -8/21)","explanation_zh":"g係數=(1/21, -8/21)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054096,"parameters":{"m":21,"b":8,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":8,\"domain\":\"reals\",\"m\":21,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0097","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-22x+(9)，m=-22≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/22, 9/22)","explanation_zh":"g係數=(-1/22, 9/22)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054097,"parameters":{"m":-22,"b":9,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":9,\"domain\":\"reals\",\"m\":-22,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0098","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=23x+(10)，m=23≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/23, -10/23)","explanation_zh":"g係數=(1/23, -10/23)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054098,"parameters":{"m":23,"b":10,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":10,\"domain\":\"reals\",\"m\":23,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0099","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=-24x+(11)，m=-24≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(-1/24, 11/24)","explanation_zh":"g係數=(-1/24, 11/24)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054099,"parameters":{"m":-24,"b":11,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":11,\"domain\":\"reals\",\"m\":-24,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r04-inverse-function-0100","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","language":"zh-Hant","source_mko_ids":["mko-inverse-function"],"title_zh":"非零斜率仿射反函數的係數","stem_zh":"在實數論域明示 f:R→R，f(x)=25x+(12)，m=25≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。","answer":{"kind":"expression","value":"(1/25, -12/25)","explanation_zh":"g係數=(1/25, -12/25)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零斜率仿射反函數的係數"]},"variant":{"family_id":"qf-inverse-affine-coefficients","seed":202610054100,"parameters":{"m":25,"b":12,"domain":"reals","operation":"affine_inverse_coefficients","semantic_template":"qf-inverse-affine-coefficients/{\"b\":12,\"domain\":\"reals\",\"m\":25,\"operation\":\"affine_inverse_coefficients\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}