# Question 1178: Affine Functions at One Input

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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.

R0001-01178 · q-20261003-r05-linear-function-0078



## Original question and givens

Over R, f(x)=(-10)x+(0) and g(x)=(-9)x+(1). At the single input x=-1, are f(x) and g(x) equal?


- G001 — Over R, f(x)=(-10)x+(0) and g(x)=(-9)x+(1). At the single input x=-1, are f(x) and g(x) equal?





## 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(-10, -1)

After

m·x = 10

Conditions: none

Substitute the same input and multiply first.

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





S003 · calculationBefore

f(x) := sum(10, 0)

After

f(x) = 10

Conditions: none

Then add the constant intercept.

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





S004 · calculationBefore

u·x := mul(-9, -1)

After

u·x = 9

Conditions: none

Evaluate the second function at the same input, not a different point.

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





S005 · calculationBefore

g(x) := sum(9, 1)

After

g(x) = 10

Conditions: none

Complete the second function's substitution.

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





S006 · impliesBefore

f(-1)=10; g(-1)=10

After

equal at this point: Yes

Conditions: none

Decide equality at this point only; agreement here does not imply global equality.

Step identifiers and dependenciesRule: pointwise_not_global · 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 — Recompute g(x) with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute u·x with exact rational arithmetic, not rounded decimals.
- 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

Yes

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: e7d96b01152bc05843015dd51a346f8897cc488f3ac93681538dcf62eb2b25e5
- candidate record SHA-256: 5d2d099ab5a72fdab49e4d29aebf925f530b542bed4aff28a450824a083916ee
- 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-0078.json)