Question 3050: Composition of Explicit Affine Expressions

Original question and givens

Let f:R→R be f(t)=(0)t+(12) and g:R→R be g(t)=(2)t+(-5). Find (g∘f)(12) exactly.

• G001 — Let f:R→R be f(t)=(0)t+(12) and g:R→R be g(t)=(2)t+(-5). Find (g∘f)(12) 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-composition

Step-by-step derivation

(g∘f)(x)

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

f(x) product := mul(0, 12)

f(x) product = 0

f(x) value := sum(0, 12)

f(x) value = 12

g(f(x)) product := mul(2, 12)

g(f(x)) product = 24

g(f(x)) value := sum(24, -5)

g(f(x)) value = 19

Reverse checks

• S005 · premise_and_result_audit — Recompute g(f(x)) value with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute g(f(x)) product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute f(x) value with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute f(x) product 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

19

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.