Question 8108: 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=-3≠0, b=-5; ordered rational inputs y=[-5/3, 8/4] 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=-3≠0, b=-5; ordered rational inputs y=[-5/3, 8/4] 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=-3;b=-5;raw ys=[[-5, 3], [8, 4]]

normalized ys=[-5/3, 2/1];a!=0

inverse slope := div(1, -3)

inverse slope = -1/3

negative intercept := sub(0, -5)

negative intercept = 5

inverse intercept := div(5, -3)

inverse intercept = -5/3

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(-5/3, -5)

query0 y-b = 10/3

query0 recovered x := div(10/3, -3)

query0 recovered x = -10/9

query0 a*x := mul(-3, -10/9)

query0 a*x = 10/3

query0 f(x) := sum(10/3, -5)

query0 f(x) = -5/3

query0:y=-5/3;x=-10/9

f(x)=-5/3;matches=Yes

query1 y-b := sub(2, -5)

query1 y-b = 7

query1 recovered x := div(7, -3)

query1 recovered x = -7/3

query1 a*x := mul(-3, -7/3)

query1 a*x = 7

query1 f(x) := sum(7, -5)

query1 f(x) = 2

query1:y=2/1;x=-7/3

f(x)=2/1;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

([-1/3, -5/3], [-10/9, -7/3], [-5/3, 2/1], 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.