Question 3097: Comparing Two Composition Orders

Original question and givens

Let f(t)=(-3)t+(9) and g(t)=(-1)t+(-2). At the explicit input x=9, are (g∘f)(x) and (f∘g)(x) equal?

• G001 — Let f(t)=(-3)t+(9) and g(t)=(-1)t+(-2). At the explicit input x=9, are (g∘f)(x) and (f∘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-composition

Step-by-step derivation

(g∘f)(x)

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

f(x) product := mul(-3, 9)

f(x) product = -27

f(x) value := sum(-27, 9)

f(x) value = -18

g(f(x)) product := mul(-1, -18)

g(f(x)) product = 18

g(f(x)) value := sum(18, -2)

g(f(x)) value = 16

g(x) product := mul(-1, 9)

g(x) product = -9

g(x) value := sum(-9, -2)

g(x) value = -11

f(g(x)) product := mul(-3, -11)

f(g(x)) product = 33

f(g(x)) value := sum(33, 9)

f(g(x)) value = 42

g(f(9))=16;f(g(9))=42

equal at supplied input=No

Reverse checks

• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute f(g(x)) value with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute f(g(x)) product with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute g(x) value with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute g(x) product with exact rational arithmetic, not rounded decimals.
• 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

No

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.