Question 1194: Affine Functions at One Input

Original question and givens

Over R, f(x)=(6)x+(1) and g(x)=(7)x+(0). At the single input x=1, are f(x) and g(x) equal?

• G001 — Over R, f(x)=(6)x+(1) and g(x)=(7)x+(0). 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

Step-by-step derivation

f:R→R; f(x)=mx+b

m=0 gives a constant affine function

m·x := mul(6, 1)

m·x = 6

f(x) := sum(6, 1)

f(x) = 7

u·x := mul(7, 1)

u·x = 7

g(x) := sum(7, 0)

g(x) = 7

f(1)=7; g(1)=7

equal at this point: Yes

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.