# Question 8110: Nonzero-Slope Real Affine Inverse and Two Rational Round Trips [Return to catalogue page 41](https://ocme.evemisslab.com/en/questions/page/41/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r02-inverse-function-foundations/#q-20261012-advance-20261017-r02-inverse-function-0085) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0015-08110 · q-20261012-advance-20261017-r02-inverse-function-0085 ## Original question and givens Declare f:R→R by f(x)=a*x+b with a=-5≠0, b=-3; ordered rational inputs y=[-3/3, 10/2] 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=-5≠0, b=-3; ordered rational inputs y=[-3/3, 10/2] 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](https://ocme.evemisslab.com/en/mko/mko-inverse-function/) ## Step-by-step derivation S001 · impliesBefore a=-5;b=-3;raw ys=[[-3, 3], [10, 2]] After normalized ys=[-1/1, 5/1];a!=0 Conditions: none For the declared affine map R→R,a nonzero slope permits solving a*x+b=y.Negative original denominators are legal and normalized first.An inverse function is not1/f(x). Step identifiers and dependenciesRule: affine_nonzero_slope_guard · Uses: G001 S002 · calculationBefore inverse slope := div(1, -5) After inverse slope = -1/5 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore negative intercept := sub(0, -3) After negative intercept = 3 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore inverse intercept := div(3, -5) After inverse intercept = -3/5 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore g(y)=(y-b)/a;f(x)=a*x+b;a!=0 After g(f(x))=((a*x+b)-b)/a=x;f(g(y))=a*((y-b)/a)+b=y Conditions: none Simplify the two composition identities separately.Nonzero a permits cancellation.The two evaluations below audit instances rather than infer a global law from two points. Step identifiers and dependenciesRule: affine_both_global_identities · Uses: G001, S004 S006 · calculationBefore query0 y-b := sub(-1, -3) After query0 y-b = 2 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore query0 recovered x := div(2, -5) After query0 recovered x = -2/5 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore query0 a*x := mul(-5, -2/5) After query0 a*x = 2 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore query0 f(x) := sum(2, -3) After query0 f(x) = -1 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore query0:y=-1/1;x=-2/5 After f(x)=-1/1;matches=Yes Conditions: none Substitute the recovered preimage into the original a and b,not into the inverse itself.Retain each of the two ordered queries. Step identifiers and dependenciesRule: affine_original_map_roundtrip · Uses: G001, S009 S011 · calculationBefore query1 y-b := sub(5, -3) After query1 y-b = 8 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore query1 recovered x := div(8, -5) After query1 recovered x = -8/5 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore query1 a*x := mul(-5, -8/5) After query1 a*x = 8 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore query1 f(x) := sum(8, -3) After query1 f(x) = 5 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore query1:y=5/1;x=-8/5 After f(x)=5/1;matches=Yes Conditions: none Substitute the recovered preimage into the original a and b,not into the inverse itself.Retain each of the two ordered queries. Step identifiers and dependenciesRule: affine_original_map_roundtrip · Uses: G001, S014 ## 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 ([-1/5, -3/5], [-2/5, -8/5], [-1/1, 5/1], 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 6d7c2b1c6a5b8f9de498b2665fd2d355837fdedd0f9f5d56c8ca60c47f6dd3da - candidate record SHA-256: f1292881e2cc65e5650c2d2d6e1a8c132949926baafd856c579b84daaaf96e33 - semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18 - question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r02-inverse-function-0085.json)