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Question 10240: Graph Membership with a Domain Restriction

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0019-10240 · q-20261003-r06-function-graph-0015

Original question and givens

In a fixed Cartesian coordinate system, D={2, 3, 4} and f:D→R is given by f(x)=(2)x+(1). Does (4, 9) belong to the declared graph?

  • G001 — In a fixed Cartesian coordinate system, D={2, 3, 4} and f:D→R is given by f(x)=(2)x+(1). Does (4, 9) belong to the declared graph?

Definition and scope

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.

Step-by-step derivation

S001 · implies

Before

raw domain=[2, 3, 4]

After

sorted distinct declared domain=[2, 3, 4]

Conditions: none

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.

Step identifiers and dependencies

Rule: function_graph_declared_domain_scope · Uses: G001

S002 · calculation

Before

queried expression m*x := mul(2, 4)

After

queried expression m*x = 8

Conditions: none

Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

queried expression m*x+b := sum(8, 1)

After

queried expression m*x+b = 9

Conditions: none

Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

candidate point=(4, 9);x in domain=Yes;ambient formula value=9

After

output equality=Yes;graph member=Yes

Conditions: none

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.

Step identifiers and dependencies

Rule: restricted_graph_membership_two_guards · Uses: G001, S003

Reverse checks

  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Recompute queried expression m*x+b with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute queried expression m*x with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

Yes

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 1b2b72e20116a4b17c90cc3d9448d18bda87340a5cef4fe0cf02d5a34e64efd9
  • candidate record SHA-256: 829a48436078d1c8e37226110eab9a48d821af4d0ad0ffd07d82494e6705d094
  • semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
  • question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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