Question 8123: 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=10; ordered rational inputs y=[10/3, 23/3] 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=10; ordered rational inputs y=[10/3, 23/3] 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=10;raw ys=[[10, 3], [23, 3]]

normalized ys=[10/3, 23/3];a!=0

inverse slope := div(1, 3)

inverse slope = 1/3

negative intercept := sub(0, 10)

negative intercept = -10

inverse intercept := div(-10, 3)

inverse intercept = -10/3

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(10/3, 10)

query0 y-b = -20/3

query0 recovered x := div(-20/3, 3)

query0 recovered x = -20/9

query0 a*x := mul(3, -20/9)

query0 a*x = -20/3

query0 f(x) := sum(-20/3, 10)

query0 f(x) = 10/3

query0:y=10/3;x=-20/9

f(x)=10/3;matches=Yes

query1 y-b := sub(23/3, 10)

query1 y-b = -7/3

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

query1 recovered x = -7/9

query1 a*x := mul(3, -7/9)

query1 a*x = -7/3

query1 f(x) := sum(-7/3, 10)

query1 f(x) = 23/3

query1:y=23/3;x=-7/9

f(x)=23/3;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, -10/3], [-20/9, -7/9], [10/3, 23/3], 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.