{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0001","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0638e30ae9ee8e6c46569e4b2eddbf7be5a88ded82d44c8f21fe984c25a3d98c","family_id":"qf-inverse-function-table","seed":202610054001,"semantic_template":"qf-inverse-function-table/{\"A\":[],\"B\":[],\"operation\":\"inverse_table\",\"pairs\":[]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A=∅, B=∅, and table ∅. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","answer":{"kind":"expression","value":"∅","explanation":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as ∅. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0002","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07f19d650c3b7df8f6b57f1380c1e6c9fa7a014d85297a7cc3b7a1f2fdcd8f8a","family_id":"qf-inverse-function-table","seed":202610054002,"semantic_template":"qf-inverse-function-table/{\"A\":[-3,-2,-1],\"B\":[1,3,5],\"operation\":\"inverse_table\",\"pairs\":[[-3,5],[-2,1],[-1,3]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A={-3, -2, -1}, B={1, 3, 5}, and table {(-3, 5), (-2, 1), (-1, 3)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","answer":{"kind":"expression","value":"{(1, -2), (3, -1), (5, -3)}","explanation":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -2), (3, -1), (5, -3)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0003","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a21171983ce736d7876be800c7a71384c831f280b49316b79f1e4f7033c3ccfa","family_id":"qf-inverse-function-table","seed":202610054003,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A={-4, -3, -2, -1}, B={1, 3, 5, 7}, and table {(-4, 7), (-3, 1), (-2, 3), (-1, 5)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","answer":{"kind":"expression","value":"{(1, -3), (3, -2), (5, -1), (7, -4)}","explanation":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -3), (3, -2), (5, -1), (7, -4)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0004","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab22b137b8efa30df118072d7c36effdb3c21ac83876776c9d32737b084f52f0","family_id":"qf-inverse-function-table","seed":202610054004,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9}, and table {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","answer":{"kind":"expression","value":"{(1, -4), (3, -3), (5, -2), (7, -1), (9, -5)}","explanation":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -4), (3, -3), (5, -2), (7, -1), (9, -5)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0005","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c43cdc1290b1de10ace32258aa36d83a14be664b4f42a7ef177cd82486239a0","family_id":"qf-inverse-function-table","seed":202610054005,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11}, and table {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","answer":{"kind":"expression","value":"{(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -6)}","explanation":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -6)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0006","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e897b12961c09f15a3a17ae357afa20b41c9dcc5325a656c9355d36758929f3b","family_id":"qf-inverse-function-table","seed":202610054006,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A={-7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-7, 13), (-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","answer":{"kind":"expression","value":"{(1, -6), (3, -5), (5, -4), (7, -3), (9, -2), (11, -1), (13, -7)}","explanation":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -6), (3, -5), (5, -4), (7, -3), (9, -2), (11, -1), (13, -7)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0007","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8db8481a5f485bc3b4d8ae72f840e108feb8cb1c398765bb86e8e0d6860d65c","family_id":"qf-inverse-function-table","seed":202610054007,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A={-8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-8, 15), (-7, 1), (-6, 3), (-5, 5), (-4, 7), (-3, 9), (-2, 11), (-1, 13)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","answer":{"kind":"expression","value":"{(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -8)}","explanation":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -8)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0008","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3ec7bf346bacb0a475c1116410c44e5e848f8befd365bb9c7d70659985c5098","family_id":"qf-inverse-function-table","seed":202610054008,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A={-9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17}, and table {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","answer":{"kind":"expression","value":"{(1, -8), (3, -7), (5, -6), (7, -5), (9, -4), (11, -3), (13, -2), (15, -1), (17, -9)}","explanation":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -8), (3, -7), (5, -6), (7, -5), (9, -4), (11, -3), (13, -2), (15, -1), (17, -9)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0009","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46d61380927b2eac3a9a3a6d2e1e26e65365746068a561696472ccff5bf194bc","family_id":"qf-inverse-function-table","seed":202610054009,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A={-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, and table {(-10, 19), (-9, 1), (-8, 3), (-7, 5), (-6, 7), (-5, 9), (-4, 11), (-3, 13), (-2, 15), (-1, 17)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -10)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0010","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3faa0acb9ae53ece0a069e0f9170455aed4c65022be05f8f9889852097a9989","family_id":"qf-inverse-function-table","seed":202610054010,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has A={-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}, and table {(-11, 21), (-10, 1), (-9, 3), (-8, 5), (-7, 7), (-6, 9), (-5, 11), (-4, 13), (-3, 15), (-2, 17), (-1, 19)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -10), (3, -9), (5, -8), (7, -7), (9, -6), (11, -5), (13, -4), (15, -3), (17, -2), (19, -1), (21, -11)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0011","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc67841b605e03cd613572f599db15d0fdd6da2e7ebe49249f59b6512cea9cdf","family_id":"qf-inverse-function-table","seed":202610054011,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-12, 23), (-11, 1), (-10, 3), (-9, 5), (-8, 7), (-7, 9), (-6, 11), (-5, 13), (-4, 15), (-3, 17), (-2, 19), (-1, 21)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -12)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0012","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96feb00c15d15585a434bffc20549cb8e5e1dd6449f6cf3f9d96cbdfcf822c88","family_id":"qf-inverse-function-table","seed":202610054012,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0013","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85b9bd36ef8c6e59576cec7284cf878945e85aaf69e0dceebfa860d88c8f6059","family_id":"qf-inverse-function-table","seed":202610054013,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0014","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61cbb0da207951a7747d7725ac24f4399684fc3351c0de9b15f242f55a614f74","family_id":"qf-inverse-function-table","seed":202610054014,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0015","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47ffb46a3257ae046ce478a41dba66e1ec7a35ff56a902aa94abc6dbddc6d885","family_id":"qf-inverse-function-table","seed":202610054015,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0016","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"741873a22eb870627419482ea5400f67fff864c59df064004ae1a6dc4d410316","family_id":"qf-inverse-function-table","seed":202610054016,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0017","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd1b8bb2ec44151466a2eff69cb8cf58beb0843e11472bade1b87dcb971f42c7","family_id":"qf-inverse-function-table","seed":202610054017,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0018","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da1ba2591e67efced1ca28464badf0a76ede48247f1e5a44db6eefdc63a88470","family_id":"qf-inverse-function-table","seed":202610054018,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0019","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73186cc4fc1afb24c7a29b578897a519fe15e6abc41d7ae00da2e45d4678fc92","family_id":"qf-inverse-function-table","seed":202610054019,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0020","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bdccc64baa9dcea6a14a9388398a7861b6472adb629ff58cdd3509debe7e2e5","family_id":"qf-inverse-function-table","seed":202610054020,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0021","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43982bf47d38cebf30130463c0800b2e0753335038b95a134b9c5e9f65dcbe77","family_id":"qf-inverse-function-table","seed":202610054021,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0022","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecac26be415c9b77b74aced26b87ff88e6f54abc286c796f18a71a2df2308eb7","family_id":"qf-inverse-function-table","seed":202610054022,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0023","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"302b08955581c1374c51d8902fa1d775551564ef21b29b91cdbe621c8e397495","family_id":"qf-inverse-function-table","seed":202610054023,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0024","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcac0eb828171127f2812b50e748a01828adb4a2d2eba26823778709aff2c001","family_id":"qf-inverse-function-table","seed":202610054024,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0025","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d5f69e867f57de2a1bc048ebe2689f3cc15a02c6549742f9d451602274425b3","family_id":"qf-inverse-function-table","seed":202610054025,"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]]}"},"fields":{"title":"Inverse Table of a Finite Bijection","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.","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":"Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(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)}. The empty bijection has the empty inverse."},"skills":["Inverse Table of a Finite Bijection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0026","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ad06b64fef2cd7a4cf078770d2780a47ff689b19ba2dad9b0b3e6db4246db4b","family_id":"qf-inverse-function-evaluation","seed":202610054026,"semantic_template":"qf-inverse-function-evaluation/{\"A\":[-2,-1],\"B\":[1,3],\"operation\":\"inverse_evaluate\",\"pairs\":[[-2,3],[-1,1]],\"t\":1}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-2, -1}, B={1, 3}, and table {(-2, 3), (-1, 1)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 1∈B, find g(1).","answer":{"kind":"short_text","value":"-1","explanation":"Read the unique input mapped by f to 1, obtaining g(1)=-1. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0027","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2db1b952302c8f7864bfce00a992d2650842bb81afefaca828af8b0e48e711c9","family_id":"qf-inverse-function-evaluation","seed":202610054027,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-3, -2, -1}, B={1, 3, 5}, and table {(-3, 5), (-2, 1), (-1, 3)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 5∈B, find g(5).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 5, obtaining g(5)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0028","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd1da5b39e993b22d4c6bece3250f4c64b233989fcba2e0739355d10cefd3133","family_id":"qf-inverse-function-evaluation","seed":202610054028,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-4, -3, -2, -1}, B={1, 3, 5, 7}, and table {(-4, 7), (-3, 1), (-2, 3), (-1, 5)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 1∈B, find g(1).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 1, obtaining g(1)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0029","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3f2fbb776c537433356fa0b05a2e24cf1edb18282ba3e19b4c725a3e6176810","family_id":"qf-inverse-function-evaluation","seed":202610054029,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9}, and table {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 3∈B, find g(3).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 3, obtaining g(3)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0030","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3795944acc03dd20e6d07843565902e6ab6c5089ba9ce334c3667b6d489f57b1","family_id":"qf-inverse-function-evaluation","seed":202610054030,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11}, and table {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 5∈B, find g(5).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 5, obtaining g(5)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0031","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcaff5b1edcd3d94a9bdd0cb6b23d7cff41ca44353f5cc1ed2c12ebf1279ab7b","family_id":"qf-inverse-function-evaluation","seed":202610054031,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-7, 13), (-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 7∈B, find g(7).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 7, obtaining g(7)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0032","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3eabd6131643f6b413774ddfb9eed1995c343d8d5972a0486b41ad3424cd936","family_id":"qf-inverse-function-evaluation","seed":202610054032,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-8, 15), (-7, 1), (-6, 3), (-5, 5), (-4, 7), (-3, 9), (-2, 11), (-1, 13)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 9∈B, find g(9).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 9, obtaining g(9)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0033","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f50007e952118e0dbf0c12036c90d1b6f4fbb71329f1d316d97b196dc2984979","family_id":"qf-inverse-function-evaluation","seed":202610054033,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17}, and table {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 11∈B, find g(11).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 11, obtaining g(11)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0034","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94a1d7e30528dd6a692501c7d41a9e805c47286b71264623bbd4e468484568bc","family_id":"qf-inverse-function-evaluation","seed":202610054034,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, and table {(-10, 19), (-9, 1), (-8, 3), (-7, 5), (-6, 7), (-5, 9), (-4, 11), (-3, 13), (-2, 15), (-1, 17)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 13∈B, find g(13).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 13, obtaining g(13)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0035","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42a4b9ecb87d2e9447e7b83a0e210043e766284afe367e7c09992471c521d852","family_id":"qf-inverse-function-evaluation","seed":202610054035,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has A={-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}, and table {(-11, 21), (-10, 1), (-9, 3), (-8, 5), (-7, 7), (-6, 9), (-5, 11), (-4, 13), (-3, 15), (-2, 17), (-1, 19)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 15∈B, find g(15).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 15, obtaining g(15)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0036","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d12723ae3bff5cc2b65b9cafd87a0eed58e9c3b2f1ba51eacb15400e89ffa202","family_id":"qf-inverse-function-evaluation","seed":202610054036,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-12, 23), (-11, 1), (-10, 3), (-9, 5), (-8, 7), (-7, 9), (-6, 11), (-5, 13), (-4, 15), (-3, 17), (-2, 19), (-1, 21)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 17∈B, find g(17).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 17, obtaining g(17)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0037","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1a1b2a1e1b14bb356e10d7c50b1e93a92435aa8f1086f6491911fbbac9ef45d","family_id":"qf-inverse-function-evaluation","seed":202610054037,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 19∈B, find g(19).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 19, obtaining g(19)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0038","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c619d972c210f2031c3944cefa2da325e174f7f46a638bdfb6302e99424933b1","family_id":"qf-inverse-function-evaluation","seed":202610054038,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 21∈B, find g(21).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 21, obtaining g(21)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0039","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32a756ee7843ec175958f804e09220d9a380702274d089106009bd0a4ec3042b","family_id":"qf-inverse-function-evaluation","seed":202610054039,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 23∈B, find g(23).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 23, obtaining g(23)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0040","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaaad038f0bc87bab3a05c1c736c7bfe211f3a21623100880854f053d0f53c0f","family_id":"qf-inverse-function-evaluation","seed":202610054040,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 25∈B, find g(25).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 25, obtaining g(25)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0041","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6b15f6fc17a8d854979570f53773f70957a5c292374c5b67460bc2e012b5a8a","family_id":"qf-inverse-function-evaluation","seed":202610054041,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 27∈B, find g(27).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 27, obtaining g(27)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0042","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2f29366cb07f9fceeb70ef9c4842b4c790dc4838723469ce532559cb415cf69","family_id":"qf-inverse-function-evaluation","seed":202610054042,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 29∈B, find g(29).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 29, obtaining g(29)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0043","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"102f07c65ce1ce819e2aa9df3545c00ecbe9b03b376908f903d340e45626a2a6","family_id":"qf-inverse-function-evaluation","seed":202610054043,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 31∈B, find g(31).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 31, obtaining g(31)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0044","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"480c9ddce85fa2c9ce9faf499c19b6c41836be13750610ea83f15e737b1c639c","family_id":"qf-inverse-function-evaluation","seed":202610054044,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 33∈B, find g(33).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 33, obtaining g(33)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0045","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03d22e1a76889ee00c67bedc81462ba4ce155d6697e486c981a22325b693eb26","family_id":"qf-inverse-function-evaluation","seed":202610054045,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 35∈B, find g(35).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 35, obtaining g(35)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0046","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07f96c6092c085de2bf7f13029d94b6e2301b7d420d24dce5c7f1a91a90eda98","family_id":"qf-inverse-function-evaluation","seed":202610054046,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 37∈B, find g(37).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 37, obtaining g(37)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0047","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37f76af1830bf295193ac3451798c77b4bdfedfd89042f42d056d2364c995138","family_id":"qf-inverse-function-evaluation","seed":202610054047,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 39∈B, find g(39).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 39, obtaining g(39)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0048","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce3965cc55a518d1fd5be2ff62843ba10e2eb6cd0f98028cd40c1b25f5f08a9a","family_id":"qf-inverse-function-evaluation","seed":202610054048,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 41∈B, find g(41).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 41, obtaining g(41)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0049","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1b70d7a7bc3446632d1726f23440a1ab6e0d2c526bc158c232d6afce00eafb5","family_id":"qf-inverse-function-evaluation","seed":202610054049,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 43∈B, find g(43).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 43, obtaining g(43)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0050","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94675711417f1618a9aa3065d843f84554620ae914f1fed5514b9bb3f366d311","family_id":"qf-inverse-function-evaluation","seed":202610054050,"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}"},"fields":{"title":"Inverse Value at a Permitted Target","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 45∈B, find g(45).","answer":{"kind":"short_text","value":"-3","explanation":"Read the unique input mapped by f to 45, obtaining g(45)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value."},"skills":["Inverse Value at a Permitted Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0051","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c52be45c5c324e301c825ec54172932aa2eda4c9f32858089008ca6a807a8ee","family_id":"qf-inverse-function-candidate","seed":202610054051,"semantic_template":"qf-inverse-function-candidate/{\"A\":[],\"B\":[],\"G\":[],\"operation\":\"inverse_candidate\",\"pairs\":[]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A=∅, B=∅, and table ∅. The function has been confirmed bijective. The candidate total g:B→A has table ∅. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0052","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f34c2e36d6e7d86580982b4cfd2c6e96f43261cf177c8bf84a08c15c0c07250","family_id":"qf-inverse-function-candidate","seed":202610054052,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A={-3, -2, -1}, B={1, 3, 5}, and table {(-3, 5), (-2, 1), (-1, 3)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -1), (3, -3), (5, -2)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0053","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"644bbf4340898f39d1e9e468a38f289d0f2e18187dd0060419c1a66e9e4018fb","family_id":"qf-inverse-function-candidate","seed":202610054053,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A={-4, -3, -2, -1}, B={1, 3, 5, 7}, and table {(-4, 7), (-3, 1), (-2, 3), (-1, 5)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -3), (3, -2), (5, -1), (7, -4)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0054","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a1d6ea41bb810f4350bebbe5178c38d9dcd3adadf74c7f9d83225f7e4228ad0","family_id":"qf-inverse-function-candidate","seed":202610054054,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9}, and table {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -3), (3, -2), (5, -1), (7, -5), (9, -4)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0055","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"053e012ac312910e5b86eed1d7a24a01b2eb2eacb252645dc9a455f135f99a64","family_id":"qf-inverse-function-candidate","seed":202610054055,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11}, and table {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -6)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0056","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60c549096ee838cdb7422f36e0bfed6e84dce1e9878cff154dabd885da0841e4","family_id":"qf-inverse-function-candidate","seed":202610054056,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A={-7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-7, 13), (-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -7), (13, -6)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0057","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf68928dcc0780a6ee2471d62964938db2e0f553a1d7bd43b844bc0182971af5","family_id":"qf-inverse-function-candidate","seed":202610054057,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A={-8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-8, 15), (-7, 1), (-6, 3), (-5, 5), (-4, 7), (-3, 9), (-2, 11), (-1, 13)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -8)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0058","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2fe53a7ef9a5885aac4b134e7632e30254104616e16589557bdcb2962b98b5c","family_id":"qf-inverse-function-candidate","seed":202610054058,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A={-9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17}, and table {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -9), (17, -8)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0059","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d86a0e178a7fba80e8f650739427ed33f26ba27feaa4f5f54358d7ff333025c3","family_id":"qf-inverse-function-candidate","seed":202610054059,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A={-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, and table {(-10, 19), (-9, 1), (-8, 3), (-7, 5), (-6, 7), (-5, 9), (-4, 11), (-3, 13), (-2, 15), (-1, 17)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -10)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0060","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"534a647360863dfc59aeb714364aa898cc4b43c92d3e1f8550be56cb0eda82d4","family_id":"qf-inverse-function-candidate","seed":202610054060,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has A={-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}, and table {(-11, 21), (-10, 1), (-9, 3), (-8, 5), (-7, 7), (-6, 9), (-5, 11), (-4, 13), (-3, 15), (-2, 17), (-1, 19)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -11), (21, -10)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0061","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd5a4d55f1233277b53d672dbef361ecfec07b6cb63582e56e27249e7b3f3347","family_id":"qf-inverse-function-candidate","seed":202610054061,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-12, 23), (-11, 1), (-10, 3), (-9, 5), (-8, 7), (-7, 9), (-6, 11), (-5, 13), (-4, 15), (-3, 17), (-2, 19), (-1, 21)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -12)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0062","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff41f84f0480c257d504e026d2427ff9ff41fcacd2dd1f03d8e27bc9ff96bdcd","family_id":"qf-inverse-function-candidate","seed":202610054062,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0063","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d934a05e0c072991c2482f3cb9be3049f2bbf79cf69ea425e94d852bf63d0e64","family_id":"qf-inverse-function-candidate","seed":202610054063,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0064","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b48aa53a6befe5b1f2c63ae64ab4f22c2e0052ff100ebdfa2ec36708bbb3102","family_id":"qf-inverse-function-candidate","seed":202610054064,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0065","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bb9641046c006645b572f9d0737d49c5a140af2846ab13b1854e386b2018a3f","family_id":"qf-inverse-function-candidate","seed":202610054065,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0066","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7f9ed3182d2970d6854ac517433ab849ca6805255ea28c54bc3f26e298cd5ad","family_id":"qf-inverse-function-candidate","seed":202610054066,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0067","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4ec4b748d93f71431e17cd47cfd6294f2f03a08fac14bd101ec135b23dc28fa","family_id":"qf-inverse-function-candidate","seed":202610054067,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0068","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"717d7344752118d6fb861a64859692011db83b3856c53cb3f3f4b45de70f35c9","family_id":"qf-inverse-function-candidate","seed":202610054068,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0069","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a5ccb5235bc296950f6ff8eda43d261c324a9678df6673c9f1ff3a4e350e084","family_id":"qf-inverse-function-candidate","seed":202610054069,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0070","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ff74744a30c426c787d6e555da4f372e96b56df5f7aaf45150c4b4779b2c4fc","family_id":"qf-inverse-function-candidate","seed":202610054070,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0071","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd7295c3051311d0a07d9c5d5860131edfe99cbe65aab9373de293888ca5fa0d","family_id":"qf-inverse-function-candidate","seed":202610054071,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0072","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1efd95ca252ebb2dc8d49cae7389733d518b4fa2e07fa8d7ff41ce7dcd34cf2a","family_id":"qf-inverse-function-candidate","seed":202610054072,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0073","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93b528de852cbf4888ca8cb06e8dad3a7a8e16ce6094509527703db091431ca9","family_id":"qf-inverse-function-candidate","seed":202610054073,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0074","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a1e828bad9c16d6b0c10d4f6f16531b5de115a79a1dde3525513dbf4ec0d454","family_id":"qf-inverse-function-candidate","seed":202610054074,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"No","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0075","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6398af771b77d7a87079f767ad9d87691505e6e1aae42cb807798f8a941d091a","family_id":"qf-inverse-function-candidate","seed":202610054075,"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]]}"},"fields":{"title":"Checking Both Inverse Identities","stem":"The declared total function f:A→B has 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}, and table {(-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)}. The function has been confirmed bijective. The candidate total g:B→A has table {(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)}. Check both identities to decide whether g is the inverse of f.","answer":{"kind":"short_text","value":"Yes","explanation":"Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure."},"skills":["Checking Both Inverse Identities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0076","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec616e1aa7015a790e8071a1891e31f895e07c781e825219aebec879bf1b3119","family_id":"qf-inverse-affine-coefficients","seed":202610054076,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-12,\"domain\":\"reals\",\"m\":1,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(1)x+(-12), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 12/1)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/1, 12/1), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0077","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f04c06500f0b0096f7b6a517fdb14f836ad59b51edbc01b95277ce24433cc83","family_id":"qf-inverse-affine-coefficients","seed":202610054077,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-11,\"domain\":\"reals\",\"m\":-2,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-2)x+(-11), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/2, -11/2)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/2, -11/2), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0078","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d90df4f260053bd2c800467db07339b9f2413e979d4deae866c699bdcaa78445","family_id":"qf-inverse-affine-coefficients","seed":202610054078,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-10,\"domain\":\"reals\",\"m\":3,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(3)x+(-10), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/3, 10/3)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/3, 10/3), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0079","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e91bd14a24a024e2380dedca606e437a2b62c6f8a4e467748f8b3de238e9ad43","family_id":"qf-inverse-affine-coefficients","seed":202610054079,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-9,\"domain\":\"reals\",\"m\":-4,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-4)x+(-9), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/4, -9/4)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/4, -9/4), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0080","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e050fc77b231509a73b75d2f6ae27101832248490dc85f70dd6bd01ea8e5dba","family_id":"qf-inverse-affine-coefficients","seed":202610054080,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-8,\"domain\":\"reals\",\"m\":5,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(5)x+(-8), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/5, 8/5)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/5, 8/5), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0081","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"976ca0ef5dbf308e773868522dbc627f8cc5045f91ff8e488d3d5f3aa8842154","family_id":"qf-inverse-affine-coefficients","seed":202610054081,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-7,\"domain\":\"reals\",\"m\":-6,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-6)x+(-7), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/6, -7/6)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/6, -7/6), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0082","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65ba50ef1b1b578d2d1139cc7ec961d79532e00aef0bc319389e12c56410fed9","family_id":"qf-inverse-affine-coefficients","seed":202610054082,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-6,\"domain\":\"reals\",\"m\":7,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(7)x+(-6), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/7, 6/7)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/7, 6/7), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0083","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87ea2659e6593c007d22e753b05a0dd0cbf2e33924706c25397affca211fcf7b","family_id":"qf-inverse-affine-coefficients","seed":202610054083,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-5,\"domain\":\"reals\",\"m\":-8,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-8)x+(-5), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/8, -5/8)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/8, -5/8), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0084","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e12405e778d5512936be515e8e62e8d0f1b8225252cd9ffb49da1a4cc44aa120","family_id":"qf-inverse-affine-coefficients","seed":202610054084,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-4,\"domain\":\"reals\",\"m\":9,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(9)x+(-4), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/9, 4/9)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/9, 4/9), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0085","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a06b5a889594a0d00684dc916f382d94a2c148da2386fc592597f53c7a74a79f","family_id":"qf-inverse-affine-coefficients","seed":202610054085,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-3,\"domain\":\"reals\",\"m\":-10,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-10)x+(-3), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/10, -3/10)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/10, -3/10), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0086","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e31b5b96cc69ffa0fcab401218c94ae311e66ebbe06ec822f362424833d12468","family_id":"qf-inverse-affine-coefficients","seed":202610054086,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-2,\"domain\":\"reals\",\"m\":11,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(11)x+(-2), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/11, 2/11)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/11, 2/11), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0087","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c1834d7454e345f98c4d1deed7c7e7588a8f3985d4a4abfdf2cc8d5bcf72fb0","family_id":"qf-inverse-affine-coefficients","seed":202610054087,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":-1,\"domain\":\"reals\",\"m\":-12,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-12)x+(-1), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/12, -1/12)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/12, -1/12), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0088","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f539a285c0642bc7c89bdab687694e891896973824764b3f58dba64223d521b6","family_id":"qf-inverse-affine-coefficients","seed":202610054088,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":0,\"domain\":\"reals\",\"m\":13,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(13)x+(0), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/13, 0/1)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/13, 0/1), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0089","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74737278a09a187cdd68d88104ffc9b0224b1eac218b96bb7dcb7760bd4474c5","family_id":"qf-inverse-affine-coefficients","seed":202610054089,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":1,\"domain\":\"reals\",\"m\":-14,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-14)x+(1), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/14, 1/14)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/14, 1/14), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0090","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db72f6fcc989f4afe5d7b15ba975e377d6c71ea96b9bb930cb72847aa8603340","family_id":"qf-inverse-affine-coefficients","seed":202610054090,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":2,\"domain\":\"reals\",\"m\":15,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(15)x+(2), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/15, -2/15)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/15, -2/15), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0091","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b40dd0477e7f2ece4a9b70b5cd59d931352024b597d338ad8db7c663e930ee4a","family_id":"qf-inverse-affine-coefficients","seed":202610054091,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":3,\"domain\":\"reals\",\"m\":-16,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-16)x+(3), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/16, 3/16)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/16, 3/16), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0092","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9a28bfa1d7db933a2e11a0a4a8423a773445fc3254037ce97a8224dab5187e6","family_id":"qf-inverse-affine-coefficients","seed":202610054092,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":4,\"domain\":\"reals\",\"m\":17,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(17)x+(4), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/17, -4/17)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/17, -4/17), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0093","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a2521bb538cbbc74389224b8e5c46cceee436f709eb8bdce579c31dfd1b2ade","family_id":"qf-inverse-affine-coefficients","seed":202610054093,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":5,\"domain\":\"reals\",\"m\":-18,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-18)x+(5), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/18, 5/18)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/18, 5/18), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0094","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"848e777d31806c4be8b59f599cdced7bf2fa4cdbc11bbaa06f64b08e045499c6","family_id":"qf-inverse-affine-coefficients","seed":202610054094,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":6,\"domain\":\"reals\",\"m\":19,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(19)x+(6), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/19, -6/19)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/19, -6/19), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0095","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb538e4c7b2796cb3e4b1d919f99c022eb181110f54093edadbbc5f9e9cad507","family_id":"qf-inverse-affine-coefficients","seed":202610054095,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":7,\"domain\":\"reals\",\"m\":-20,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-20)x+(7), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/20, 7/20)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/20, 7/20), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0096","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73fc9d950c3db88d9d25d933c9e9f068428668f73350f879fc07548a0afa7f1a","family_id":"qf-inverse-affine-coefficients","seed":202610054096,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":8,\"domain\":\"reals\",\"m\":21,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(21)x+(8), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/21, -8/21)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/21, -8/21), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0097","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f72acc27583f73c394fa47059b85b77026f0ef520ddd54b40239e881287f2de2","family_id":"qf-inverse-affine-coefficients","seed":202610054097,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":9,\"domain\":\"reals\",\"m\":-22,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-22)x+(9), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/22, 9/22)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/22, 9/22), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0098","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8956c491c8561fc354b5336e415e9b89db5aab242a74d8783fc633ca82455c52","family_id":"qf-inverse-affine-coefficients","seed":202610054098,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":10,\"domain\":\"reals\",\"m\":23,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(23)x+(10), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/23, -10/23)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/23, -10/23), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0099","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f04847945b6e85f030452648ba3ada2188fbba15fa79c4edf3260c0ac9684e75","family_id":"qf-inverse-affine-coefficients","seed":202610054099,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":11,\"domain\":\"reals\",\"m\":-24,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(-24)x+(11), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/24, 11/24)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/24, 11/24), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r04-inverse-function-0100","batch_id":"batch-20261004-advance-20261005-r04-inverse-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ffd136116f6ebfacb3cc0f4b272ead6e3198a14f68457d311f34f06482da153","family_id":"qf-inverse-affine-coefficients","seed":202610054100,"semantic_template":"qf-inverse-affine-coefficients/{\"b\":12,\"domain\":\"reals\",\"m\":25,\"operation\":\"affine_inverse_coefficients\"}"},"fields":{"title":"Inverse Coefficients of a Nonzero-Slope Affine Function","stem":"Over R, let f:R→R be f(x)=(25)x+(12), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/25, -12/25)","explanation":"Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/25, -12/25), with nonzero slope ensuring invertibility in this real affine setting."},"skills":["Inverse Coefficients of a Nonzero-Slope Affine Function"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}