Question 1161: Complete Zero Set of an Affine Function

Original question and givens

Over R, f(x)=(-2)x+(-2). Find its complete zero set, using R, ∅, or a set of reduced rational numbers.

• G001 — Over R, f(x)=(-2)x+(-2). Find its complete zero set, using R, ∅, or a set of reduced rational numbers.

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

-2x+(-2)=0

m≠0: Yes

root := div(2, -2)

root = -1

m·root := mul(-2, -1)

m·root = 2

f(root) := sum(2, -2)

f(root) = 0

m(x₁−x₂)=0 with m≠0

x₁=x₂

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 f(root) with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute m·root with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute root with exact rational arithmetic, not rounded decimals.
• 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

{-1/1}

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.