# Question 3107: Composition Value from Finite Function Tables

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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.

R0005-03107 · q-20260930-r04-composition-table-evaluation-0007



## Original question and givens

The explicit total functions have A={-6, -5, -4}, B={-8, 0, 9}, C={-13, 0, 16}, f=[(-6, -8), (-5, 0), (-4, 9)], and g=[(-8, -13), (0, 16), (9, 0)]. Find (g∘f)(-6).


- G001 — The explicit total functions have A={-6, -5, -4}, B={-8, 0, 9}, C={-13, 0, 16}, f=[(-6, -8), (-5, 0), (-4, 9)], and g=[(-8, -13), (0, 16), (9, 0)]. Find (g∘f)(-6).





## 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-composition](https://ocme.evemisslab.com/en/mko/mko-composition/)





## Step-by-step derivation

S001 · impliesBefore

(g∘f)(x)

After

g(f(x)):apply f first,then g

Conditions: none

The function on the right acts first;composition is not multiplication of two function values.

Step identifiers and dependenciesRule: composition_order · Uses: G001





S002 · impliesBefore

f:A→B;g:D→C;D={-8, 0, 9}

After

both tables are total on their declared domains

Conditions: none

Audit totality and codomain membership for each table;g's domain D need not equal f's codomain B.

Step identifiers and dependenciesRule: composition_totality_guard · Uses: G001, S001





S003 · impliesBefore

x=-6;f(x)=-8;D={-8, 0, 9}

After

f(x)∈D=Yes

Conditions: none

An input is permitted exactly when its intermediate f-value lies in g's domain;membership of x in B is not the criterion.

Step identifiers and dependenciesRule: composition_domain_row · Uses: G001, S002





S004 · impliesBefore

x=-5;f(x)=0;D={-8, 0, 9}

After

f(x)∈D=Yes

Conditions: none

An input is permitted exactly when its intermediate f-value lies in g's domain;membership of x in B is not the criterion.

Step identifiers and dependenciesRule: composition_domain_row · Uses: G001, S003





S005 · impliesBefore

x=-4;f(x)=9;D={-8, 0, 9}

After

f(x)∈D=Yes

Conditions: none

An input is permitted exactly when its intermediate f-value lies in g's domain;membership of x in B is not the criterion.

Step identifiers and dependenciesRule: composition_domain_row · Uses: G001, S004





S006 · impliesBefore

x=-6∈A;f(-6)=-8∈D

After

g(-8)=-13∈C

Conditions: none

Look up f at x,then g at the intermediate value;do not interchange the lookup keys.

Step identifiers and dependenciesRule: composition_table_lookup · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

-13

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: ffbb914b39e2543ed71221196d16bac77238ecbcd8f82acf4ca7fdad69a6a876
- candidate record SHA-256: 450d088385a3b75291d26157ef5761ab40987f4cfb94cae611fefdd89f792fd0
- semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2
- question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260930-r04-composition-table-evaluation-0007.json)