Question 3058: Permitted Domain of a Restricted Composition

Original question and givens

Let f:A→B and g:D→C be the explicit total functions A={-5, -4, -3}, B={-9, 0, 10}, D={106}, C={-14, 1, 15}, f=[(-5, 0), (-4, 10), (-3, -9)], and g=[(106, 1)]. List {x∈A:f(x)∈D}.

• G001 — Let f:A→B and g:D→C be the explicit total functions A={-5, -4, -3}, B={-9, 0, 10}, D={106}, C={-14, 1, 15}, f=[(-5, 0), (-4, 10), (-3, -9)], and g=[(106, 1)]. List {x∈A:f(x)∈D}.

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:A→B;g:D→C;D={106}

both tables are total on their declared domains

x=-5;f(x)=0;D={106}

f(x)∈D=No

x=-4;f(x)=10;D={106}

f(x)∈D=No

x=-3;f(x)=-9;D={106}

f(x)∈D=No

{x∈A:f(x)∈D}

∅

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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

∅

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.