# Question 10210: Listing a Finite Function Graph [Return to catalogue page 52](https://ocme.evemisslab.com/en/questions/page/52/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-r06-function-graph-foundations/#q-20261003-r06-function-graph-0060) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0019-10210 · q-20261003-r06-function-graph-0060 ## Original question and givens In a fixed Cartesian coordinate system, D={-3, -2, -1} and f:D→R is given by f(x)=(2)x+(0). List the complete graph in increasing order of first coordinate, listing identical points once. - G001 — In a fixed Cartesian coordinate system, D={-3, -2, -1} and f:D→R is given by f(x)=(2)x+(0). List the complete graph in increasing order of first coordinate, listing identical points once. ## 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. - [mko-function-graph](https://ocme.evemisslab.com/en/mko/mko-function-graph/) ## Step-by-step derivation S001 · impliesBefore raw domain=[-3, -2, -1] After sorted distinct declared domain=[-3, -2, -1] 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 dependenciesRule: function_graph_declared_domain_scope · Uses: G001 S002 · calculationBefore domain input-3 m*x := mul(2, -3) After domain input-3 m*x = -6 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore domain input-3 m*x+b := sum(-6, 0) After domain input-3 m*x+b = -6 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore legal input=-3;formula value=-6 After graph point=(-3, -6) Conditions: none Evaluate once per legal input and list its ordered point without extending the graph outside its declared domain. Step identifiers and dependenciesRule: each_legal_affine_graph_point · Uses: G001, S003 S005 · calculationBefore domain input-2 m*x := mul(2, -2) After domain input-2 m*x = -4 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore domain input-2 m*x+b := sum(-4, 0) After domain input-2 m*x+b = -4 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore legal input=-2;formula value=-4 After graph point=(-2, -4) Conditions: none Evaluate once per legal input and list its ordered point without extending the graph outside its declared domain. Step identifiers and dependenciesRule: each_legal_affine_graph_point · Uses: G001, S006 S008 · calculationBefore domain input-1 m*x := mul(2, -1) After domain input-1 m*x = -2 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore domain input-1 m*x+b := sum(-2, 0) After domain input-1 m*x+b = -2 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore legal input=-1;formula value=-2 After graph point=(-1, -2) Conditions: none Evaluate once per legal input and list its ordered point without extending the graph outside its declared domain. Step identifiers and dependenciesRule: each_legal_affine_graph_point · Uses: G001, S009 S011 · impliesBefore complete legal-input points=[[-3, -6], [-2, -4], [-1, -2]] After finite graph={(-3, -6), (-2, -4), (-1, -2)} Conditions: none List the complete set in increasing first-coordinate order.An empty domain gives∅,not an invented(0,0). Step identifiers and dependenciesRule: complete_finite_graph_set · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute domain input-1 m*x+b with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute domain input-1 m*x with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute domain input-2 m*x+b with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute domain input-2 m*x with exact rational arithmetic, not rounded decimals. - 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 domain input-3 m*x+b with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute domain input-3 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 {(-3, -6), (-2, -4), (-1, -2)} 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: 68929c27ab077b71a55c1df854fcf872dfbcc4839398fd89b148d104a496b117 - candidate record SHA-256: 9f534014f1f5e9361574dc2cb2b992098a2ddfcb01fa9d0ec89e612533f4278f - semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5 - question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r06-function-graph-0060.json)