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R0019-10246 · q-20261003-r06-function-graph-0021
In a fixed Cartesian coordinate system, D={8, 9, 10} and f:D→R is given by f(x)=(-2)x+(-1). Does (10, -21) belong to the declared graph?
G001 — In a fixed Cartesian coordinate system, D={8, 9, 10} and f:D→R is given by f(x)=(-2)x+(-1). Does (10, -21) belong to the declared graph?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.
Before
raw domain=[8, 9, 10]
After
sorted distinct declared domain=[8, 9, 10]
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.
Rule: function_graph_declared_domain_scope · Uses: G001
Before
queried expression m*x := mul(-2, 10)
After
queried expression m*x = -20
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
queried expression m*x+b := sum(-20, -1)
After
queried expression m*x+b = -21
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
candidate point=(10, -21);x in domain=Yes;ambient formula value=-21
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.
Rule: restricted_graph_membership_two_guards · Uses: G001, S003
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.Yes
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 1d2a58c9ca2ecf27959f5bc732d3836a0383ca4f771023dc15c8765ddcd4ad72candidate record SHA-256: 7229b9c124c98f272433f7c033a0d26237df99314ab3f04a3ffeb4eb77ed6bc0semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED