# Question 32724: Inverse Coefficients of a Nonzero-Slope Affine Function [Return to catalogue page 164](https://ocme.evemisslab.com/en/questions/page/164/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261004-advance-20261005-r04-inverse-function-foundations/#q-20261004-advance-20261005-r04-inverse-function-0099) 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. R0064-32724 · q-20261004-advance-20261005-r04-inverse-function-0099 ## Original question and givens Over R, let f:R→R be f(x)=(-24)x+(11), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers. - G001 — Over R, let f:R→R be f(x)=(-24)x+(11), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers. ## 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 source roles = [qf-inverse-affine-coefficients, affine_inverse_coefficients, reals, []] After derived = [[SLOPE_FIRST_pair_fractions_RETAIN1]] Conditions: none Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status. Step identifiers and dependenciesRule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001 S002 · calculationBefore Affine inverse ORIGINAL reciprocal nonzero slope := div(1, -24) After Affine inverse ORIGINAL reciprocal nonzero slope = -1/24 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Affine inverse ORIGINAL negative constant := sub(0, 11) After Affine inverse ORIGINAL negative constant = -11 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Affine inverse ORIGINAL negative b over m := div(-11, -24) After Affine inverse ORIGINAL negative b over m = 11/24 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Affine inverse reverse g after f slope := mul(-1/24, -24) After Affine inverse reverse g after f slope = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Affine inverse reverse u times original b := mul(-1/24, 11) After Affine inverse reverse u times original b = -11/24 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Affine inverse reverse g after f constant := sum(-11/24, 11/24) After Affine inverse reverse g after f constant = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Affine inverse reverse f after g slope := mul(-24, -1/24) After Affine inverse reverse f after g slope = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Affine inverse reverse m times inverse constant := mul(-24, 11/24) After Affine inverse reverse m times inverse constant = -11 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Affine inverse reverse f after g constant := sum(-11, 11) After Affine inverse reverse f after g constant = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [-24, 11] After derived = [-1/24, 11/24, 1, 0, 1, 0] Conditions: none ONLY m≠0 permits x=(y−b)/m.The pair for g(y)=uy+v is[SLOPE,constant].BOTH whole compositions restore slope1/constant0 on ALL R.Negative m is legal;fractions retain/1. Step identifiers and dependenciesRule: Inverse_AFFINE_real_whole_law_SLOPE_FIRST_BOTH_compositions · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S010 · premise_and_result_audit — Recompute Affine inverse reverse f after g constant with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Affine inverse reverse m times inverse constant with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Affine inverse reverse f after g slope with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Affine inverse reverse g after f constant with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Affine inverse reverse u times original b with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Affine inverse reverse g after f slope with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Affine inverse ORIGINAL negative b over m with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Affine inverse ORIGINAL negative constant with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Affine inverse ORIGINAL reciprocal nonzero slope with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. ## Result (-1/24, 11/24) 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: f04847945b6e85f030452648ba3ada2188fbba15fa79c4edf3260c0ac9684e75 - candidate record SHA-256: fc1006ee5156eafedf8a1103224f814680f8695584760e0e67ee795c872aa734 - semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b - question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261004-advance-20261005-r04-inverse-function-0099.json)