# Question 10201: Listing a Finite Function Graph

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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.

R0019-10201 · q-20261003-r06-function-graph-0051



## Original question and givens

In a fixed Cartesian coordinate system, D=∅ and f:D→R is given by f(x)=(-2)x+(-1). List the complete graph in increasing order of first coordinate, listing identical points once.


- G001 — In a fixed Cartesian coordinate system, D=∅ and f:D→R is given by f(x)=(-2)x+(-1). 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=[]

After

sorted distinct declared domain=[]

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 · impliesBefore

complete legal-input points=[]

After

finite graph=∅

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, S001









## Reverse checks


- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

∅

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: b15e00f29074a9839f44d10bb91ac7c16d3fe57c12ccd7a586a8d7ae6cf47dd7
- candidate record SHA-256: 1babaa05031908ff50f74357c4985b86687509d27bc32276abfc22cc86e9135d
- 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-0051.json)