Question 8121: Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Original question and givens

Declare f:R→R by f(x)=a*x+b with a=1≠0, b=8; ordered rational inputs y=[8/3, 21/1] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

• G001 — Declare f:R→R by f(x)=a*x+b with a=1≠0, b=8; ordered rational inputs y=[8/3, 21/1] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

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-inverse-function

Step-by-step derivation

a=1;b=8;raw ys=[[8, 3], [21, 1]]

normalized ys=[8/3, 21/1];a!=0

inverse slope := div(1, 1)

inverse slope = 1

negative intercept := sub(0, 8)

negative intercept = -8

inverse intercept := div(-8, 1)

inverse intercept = -8

g(y)=(y-b)/a;f(x)=a*x+b;a!=0

g(f(x))=((a*x+b)-b)/a=x;f(g(y))=a*((y-b)/a)+b=y

query0 y-b := sub(8/3, 8)

query0 y-b = -16/3

query0 recovered x := div(-16/3, 1)

query0 recovered x = -16/3

query0 a*x := mul(1, -16/3)

query0 a*x = -16/3

query0 f(x) := sum(-16/3, 8)

query0 f(x) = 8/3

query0:y=8/3;x=-16/3

f(x)=8/3;matches=Yes

query1 y-b := sub(21, 8)

query1 y-b = 13

query1 recovered x := div(13, 1)

query1 recovered x = 13

query1 a*x := mul(1, 13)

query1 a*x = 13

query1 f(x) := sum(13, 8)

query1 f(x) = 21

query1:y=21/1;x=13/1

f(x)=21/1;matches=Yes

Reverse checks

• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute query1 f(x) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute query1 a*x with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute query1 recovered x with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute query1 y-b with exact rational arithmetic, not rounded decimals.
• 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 query0 f(x) with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute query0 a*x with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute query0 recovered x with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute query0 y-b with exact rational arithmetic, not rounded decimals.
• 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 — Recompute inverse intercept with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute negative intercept with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute inverse slope 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

([1/1, -8/1], [-16/3, 13/1], [8/3, 21/1], Yes)

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.