{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261017-r02-inverse-function-0097",
  "pilot_number": "R0015-08122",
  "number": 8122,
  "family_id": "qf-density17-inverse-affine-recovery",
  "title": {
    "zh-Hant": "第 8122 題：非零斜率的精確兩點還原",
    "en": "Question 8122: Nonzero-Slope Real Affine Inverse and Two Rational Round Trips"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261017-r02-inverse-function-0097",
    "batch_id": "batch-20261012-advance-20261017-r02-inverse-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-inverse-function"
    ],
    "title_zh": "非零斜率的精確兩點還原",
    "stem_zh": "R到R仿射模型，斜率非零。f(x)=a*x+b，a=-2、b=9，有序兩目標y=[(9)/(3), (22)/(2)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。",
    "answer": {
      "kind": "expression",
      "value": "([-1/2, 9/2], [3/1, -1/1], [3/1, 11/1], 是)",
      "explanation_zh": "離散結構核對=([-1/2, 9/2], [3/1, -1/1], [3/1, 11/1], 是)；先解a*x+b=y，再以原f精確回代；每個分數化為正分母既約值。；有限精確核對不是一般形式證明；反函數不是函數值倒數；兩側identity分開；有理答案最簡正分母。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "非零斜率的精確兩點還原"
      ]
    },
    "variant": {
      "family_id": "qf-density17-inverse-affine-recovery",
      "seed": 202610172097,
      "parameters": {
        "a": -2,
        "b": 9,
        "ys": [
          [
            9,
            3
          ],
          [
            22,
            2
          ]
        ],
        "domain": "R_affine_nonzero_integer_slice",
        "operation": "inverse-affine-recovery",
        "semantic_template": "qf-density17-inverse-affine-recovery/{\"a\":-2,\"b\":9,\"ys\":[\"3/1\",\"11/1\"]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261017-r02-inverse-function-0097",
    "batch_id": "batch-20261012-advance-20261017-r02-inverse-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "121cab84c5e8ad7db0e76efefc9a17a933cf9e72fad771e6e77a1008ffe4607e",
      "family_id": "qf-density17-inverse-affine-recovery",
      "seed": 202610172097,
      "semantic_template": "qf-density17-inverse-affine-recovery/{\"a\":-2,\"b\":9,\"ys\":[\"3/1\",\"11/1\"]}"
    },
    "fields": {
      "title": "Nonzero-Slope Real Affine Inverse and Two Rational Round Trips",
      "stem": "Declare f:R→R by f(x)=a*x+b with a=-2≠0, b=9; ordered rational inputs y=[9/3, 22/2] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.",
      "answer": {
        "kind": "expression",
        "value": "([-1/2, 9/2], [3/1, -1/1], [3/1, 11/1], Yes)",
        "explanation": "Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 9/2], [3/1, -1/1], [3/1, 11/1], Yes)."
      },
      "skills": [
        "Nonzero-Slope Real Affine Inverse and Two Rational Round Trips"
      ],
      "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": "R到R仿射模型，斜率非零。f(x)=a*x+b，a=-2、b=9，有序兩目標y=[(9)/(3), (22)/(2)]。答 ([g的一次係數, 常數], 兩個原像x, 兩個回代f(x), 兩回代都等於原y資格)。",
        "en": "Declare f:R→R by f(x)=a*x+b with a=-2≠0, b=9; ordered rational inputs y=[9/3, 22/2] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "a=-2;b=9;raw ys=[[9, 3], [22, 2]]",
      "after": "normalized ys=[3/1, 11/1];a!=0",
      "relation": "implies",
      "rule_id": "affine_nonzero_slope_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "R到R的明示仿射模型且斜率非零，才可由a*x+b=y反解；負原分母合法並先正規化。反函數不是1/f(x)。",
        "en": "For the declared affine map R→R,a nonzero slope permits solving a*x+b=y.Negative original denominators are legal and normalized first.An inverse function is not1/f(x)."
      },
      "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": "inverse slope := div(1, -2)",
      "after": "inverse slope = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inverse slope，不以小數近似替代。",
          "en": "Recompute inverse slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-1/2",
        "label": "inverse slope"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "negative intercept := sub(0, 9)",
      "after": "negative intercept = -9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 negative intercept，不以小數近似替代。",
          "en": "Recompute negative intercept with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "9"
        ],
        "output": "-9",
        "label": "negative intercept"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "inverse intercept := div(-9, -2)",
      "after": "inverse intercept = 9/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inverse intercept，不以小數近似替代。",
          "en": "Recompute inverse intercept with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-9",
          "-2"
        ],
        "output": "9/2",
        "label": "inverse intercept"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "g(y)=(y-b)/a;f(x)=a*x+b;a!=0",
      "after": "g(f(x))=((a*x+b)-b)/a=x;f(g(y))=a*((y-b)/a)+b=y",
      "relation": "implies",
      "rule_id": "affine_both_global_identities",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分別化簡兩個方向的恆等式，非零a是消去除法的條件；下面兩點只是代值核對，不是由兩點推出全域。",
        "en": "Simplify the two composition identities separately.Nonzero a permits cancellation.The two evaluations below audit instances rather than infer a global law from two points."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "query0 y-b := sub(3, 9)",
      "after": "query0 y-b = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 y-b，不以小數近似替代。",
          "en": "Recompute query0 y-b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "9"
        ],
        "output": "-6",
        "label": "query0 y-b"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "query0 recovered x := div(-6, -2)",
      "after": "query0 recovered x = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 recovered x，不以小數近似替代。",
          "en": "Recompute query0 recovered x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-6",
          "-2"
        ],
        "output": "3",
        "label": "query0 recovered x"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "query0 a*x := mul(-2, 3)",
      "after": "query0 a*x = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 a*x，不以小數近似替代。",
          "en": "Recompute query0 a*x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "3"
        ],
        "output": "-6",
        "label": "query0 a*x"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "query0 f(x) := sum(-6, 9)",
      "after": "query0 f(x) = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query0 f(x)，不以小數近似替代。",
          "en": "Recompute query0 f(x) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "9"
        ],
        "output": "3",
        "label": "query0 f(x)"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "query0:y=3/1;x=3/1",
      "after": "f(x)=3/1;matches=是",
      "relation": "implies",
      "rule_id": "affine_original_map_roundtrip",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把求得原像代回原本的a與b，不用逆函數再次檢查自己；有序兩個查詢分別保留。",
        "en": "Substitute the recovered preimage into the original a and b,not into the inverse itself.Retain each of the two ordered queries."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "query1 y-b := sub(11, 9)",
      "after": "query1 y-b = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 y-b，不以小數近似替代。",
          "en": "Recompute query1 y-b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "9"
        ],
        "output": "2",
        "label": "query1 y-b"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "query1 recovered x := div(2, -2)",
      "after": "query1 recovered x = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 recovered x，不以小數近似替代。",
          "en": "Recompute query1 recovered x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "-2"
        ],
        "output": "-1",
        "label": "query1 recovered x"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "query1 a*x := mul(-2, -1)",
      "after": "query1 a*x = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 a*x，不以小數近似替代。",
          "en": "Recompute query1 a*x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-1"
        ],
        "output": "2",
        "label": "query1 a*x"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "query1 f(x) := sum(2, 9)",
      "after": "query1 f(x) = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 query1 f(x)，不以小數近似替代。",
          "en": "Recompute query1 f(x) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "9"
        ],
        "output": "11",
        "label": "query1 f(x)"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "query1:y=11/1;x=-1/1",
      "after": "f(x)=11/1;matches=是",
      "relation": "implies",
      "rule_id": "affine_original_map_roundtrip",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把求得原像代回原本的a與b，不用逆函數再次檢查自己；有序兩個查詢分別保留。",
        "en": "Substitute the recovered preimage into the original a and b,not into the inverse itself.Retain each of the two ordered queries."
      },
      "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": [
      [
        "-1/2",
        "9/2"
      ],
      [
        "3/1",
        "-1/1"
      ],
      [
        "3/1",
        "11/1"
      ],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([-1/2, 9/2], [3/1, -1/1], [3/1, 11/1], 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": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
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      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261017-r02-inverse-function-foundations.en.jsonl"
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      "formalization": "not_formalized",
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      "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."
  },
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    "id": "review-20261026-005-p16",
    "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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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}