# Question 1124: Exact Affine-Function Evaluation

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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.

R0001-01124 · q-20261003-r05-linear-function-0024



## Original question and givens

Over R, f(x)=(11)x+(-1). Compute f(1) exactly.


- G001 — Over R, f(x)=(11)x+(-1). Compute f(1) exactly.





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





## Step-by-step derivation

S001 · impliesBefore

f:R→R; f(x)=mx+b

After

m=0 gives a constant affine function

Conditions: none

An affine function allows m=0; do not impose the nondegenerate-equation condition on every affine function.

Step identifiers and dependenciesRule: affine_domain · Uses: G001





S002 · calculationBefore

m·x := mul(11, 1)

After

m·x = 11

Conditions: none

Substitute the same input and multiply first.

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





S003 · calculationBefore

f(x) := sum(11, -1)

After

f(x) = 10

Conditions: none

Then add the constant intercept.

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





S004 · impliesBefore

x=1; m=11; b=-1

After

f(1)=10

Conditions: none

The calculation follows the original expression, retaining parentheses for negative coefficients.

Step identifiers and dependenciesRule: substitution_identity · 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 f(x) with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute 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

10

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: 77f19afe8919f2cdc81ff19c4d9c88b2f3bd8d49346c8cfc5337ff34eef29c3d
- candidate record SHA-256: cbd52bbcb146ca17b4fec231d73030ec3aa4e5362d342c651c337ba6016d0f38
- semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc
- question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r05-linear-function-0024.json)