Question 3061: Permitted Domain of a Restricted Composition

Original question and givens

Let f:A→B and g:D→C be the explicit total functions A={-2, -1, 0}, B={-12, 0, 13}, D={-12, 0}, C={-20, 1, 21}, f=[(-2, 0), (-1, 13), (0, -12)], and g=[(-12, 1), (0, 21)]. List {x∈A:f(x)∈D}.

• G001 — Let f:A→B and g:D→C be the explicit total functions A={-2, -1, 0}, B={-12, 0, 13}, D={-12, 0}, C={-20, 1, 21}, f=[(-2, 0), (-1, 13), (0, -12)], and g=[(-12, 1), (0, 21)]. 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={-12, 0}

both tables are total on their declared domains

x=-2;f(x)=0;D={-12, 0}

f(x)∈D=Yes

x=-1;f(x)=13;D={-12, 0}

f(x)∈D=No

x=0;f(x)=-12;D={-12, 0}

f(x)∈D=Yes

{x∈A:f(x)∈D}

{-2, 0}

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

{-2, 0}

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.