{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r06-function-graph-0003",
  "pilot_number": "R0019-10228",
  "number": 10228,
  "family_id": "qf-function-graph-membership",
  "title": {
    "zh-Hant": "第 10228 題：含定義域限制的圖形成員",
    "en": "Question 10228: Graph Membership with a Domain Restriction"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r06-function-graph-0003",
    "batch_id": "batch-20261003-r06-function-graph-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-function-graph"
    ],
    "title_zh": "含定義域限制的圖形成員",
    "stem_zh": "固定直角座標系，定義域 D={-10, -9, -8}，f:D→R，f(x)=(0)x+(1)。點 (-8, 1) 是否屬於宣告圖形？",
    "answer": {
      "kind": "short_text",
      "value": "是",
      "explanation_zh": "x∈D=是；mx+b=1。需同時通過定義域與輸出等式。域外代數式值不是宣告函數在域外的值。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "含定義域限制的圖形成員"
      ]
    },
    "variant": {
      "family_id": "qf-function-graph-membership",
      "seed": 202610036003,
      "parameters": {
        "A": [
          -10,
          -9,
          -8
        ],
        "m": 0,
        "b": 1,
        "domain": "reals",
        "operation": "graph_membership",
        "x": -8,
        "y": 1,
        "semantic_template": "qf-function-graph-membership/{\"A\":[-10,-9,-8],\"b\":1,\"domain\":\"reals\",\"m\":0,\"operation\":\"graph_membership\",\"x\":-8,\"y\":1}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-r06-function-graph-0003",
    "batch_id": "batch-20261003-r06-function-graph-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "fc6712ac23b50e9091c767f040cd5fbb3cd2b85b469de38ef3506f2101040130",
      "family_id": "qf-function-graph-membership",
      "seed": 202610036003,
      "semantic_template": "qf-function-graph-membership/{\"A\":[-10,-9,-8],\"b\":1,\"domain\":\"reals\",\"m\":0,\"operation\":\"graph_membership\",\"x\":-8,\"y\":1}"
    },
    "fields": {
      "title": "Graph Membership with a Domain Restriction",
      "stem": "In a fixed Cartesian coordinate system, D={-10, -9, -8} and f:D→R is given by f(x)=(0)x+(1). Does (-8, 1) belong to the declared graph?",
      "answer": {
        "kind": "short_text",
        "value": "Yes",
        "explanation": "The input -8 belongs to D, and the formula value is 1. Both the domain and output conditions must hold, giving Yes. An algebraic value outside D is not a value of the declared function there."
      },
      "skills": [
        "Graph Membership with a Domain Restriction"
      ],
      "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."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "固定直角座標系，定義域 D={-10, -9, -8}，f:D→R，f(x)=(0)x+(1)。點 (-8, 1) 是否屬於宣告圖形？",
        "en": "In a fixed Cartesian coordinate system, D={-10, -9, -8} and f:D→R is given by f(x)=(0)x+(1). Does (-8, 1) belong to the declared graph?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw domain=[-10, -9, -8]",
      "after": "sorted distinct declared domain=[-10, -9, -8]",
      "relation": "implies",
      "rule_id": "function_graph_declared_domain_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題定義域是明示有限整數子集，不自動變成全R；輸入清單須本來相異。座標第一個是輸入，第二個是輸出。",
        "en": "The domain is the declared finite integer subset,not automatically all R.The input list must already be distinct.The first coordinate is input,the second output."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "queried expression m*x := mul(0, -8)",
      "after": "queried expression m*x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 queried expression m*x，不以小數近似替代。",
          "en": "Recompute queried expression m*x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-8"
        ],
        "output": "0",
        "label": "queried expression m*x"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "queried expression m*x+b := sum(0, 1)",
      "after": "queried expression m*x+b = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 queried expression m*x+b，不以小數近似替代。",
          "en": "Recompute queried expression m*x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "queried expression m*x+b"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "candidate point=(-8, 1);x in domain=是;ambient formula value=1",
      "after": "output equality=是;graph member=是",
      "relation": "implies",
      "rule_id": "restricted_graph_membership_two_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代數表達式在域外可有數值，仍不是受限函數在域外的值；定義域與y=f(x)必須同時通過。",
        "en": "The algebraic expression can have a numerical value outside D,but that is not a value of the restricted function there.Both domain membership and y=f(x) must hold."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": true
  },
  "conclusion": {
    "canonical_value": "是",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
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    "question_binding": {
      "batch_id": "batch-20261003-r06-function-graph-foundations",
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      "batch_sha256": "672c24a566f33e7b609e784d83ad5359ac74260f1462edb0d78ab9f52ada37d0",
      "original_line": 3,
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      "question_compact_sha256": "fc6712ac23b50e9091c767f040cd5fbb3cd2b85b469de38ef3506f2101040130",
      "english_row_compact_sha256": "5db6ce5a16b4ab772300343624cf6c8c1356e82ee34af8db1bb740b97da38842",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-r06-function-graph-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-r06-function-graph-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-function-graph": {
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        "english_sha256": "4c1bb95586ab87e347b417ee2f0798ef6a6326dd0c2100e111a3f89dd223f2f2",
        "original_path": "public/data/mko/function-graph.json",
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "7359cb2346bdb6bf38b7ca4a5fd5b5a99c3007a711f34346754d57b77b2dfcc8",
    "original_candidate_sha256": "db5cb2219cfea5db2756bff8e0870d736d88f9b68132426b5085f24dcb748fa9",
    "original_candidate_line_sha256": "db5cb2219cfea5db2756bff8e0870d736d88f9b68132426b5085f24dcb748fa9",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
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      }
    }
  },
  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-006-p20",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
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      "question_first_witness_sha256": "3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
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    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}