# Question 3087: Comparing Two Composition Orders [Return to catalogue page 16](https://ocme.evemisslab.com/en/questions/page/16/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260930-r04-composition-foundations/#q-20260930-r04-composition-order-comparison-0012) 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. R0005-03087 · q-20260930-r04-composition-order-comparison-0012 ## Original question and givens Let f(t)=(1)t+(-1) and g(t)=(-1)t+(-5). At the explicit input x=-1, are (g∘f)(x) and (f∘g)(x) equal? - G001 — Let f(t)=(1)t+(-1) and g(t)=(-1)t+(-5). At the explicit input x=-1, are (g∘f)(x) and (f∘g)(x) equal? ## 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-composition](https://ocme.evemisslab.com/en/mko/mko-composition/) ## Step-by-step derivation S001 · impliesBefore (g∘f)(x) After g(f(x)):apply f first,then g Conditions: none The function on the right acts first;composition is not multiplication of two function values. Step identifiers and dependenciesRule: composition_order · Uses: G001 S002 · calculationBefore f(x) product := mul(1, -1) After f(x) product = -1 Conditions: none Substitute the current input and multiply by the coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore f(x) value := sum(-1, -1) After f(x) value = -2 Conditions: none Then add this function's own constant term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore g(f(x)) product := mul(-1, -2) After g(f(x)) product = 2 Conditions: none Substitute the current input and multiply by the coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore g(f(x)) value := sum(2, -5) After g(f(x)) value = -3 Conditions: none Then add this function's own constant term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore g(x) product := mul(-1, -1) After g(x) product = 1 Conditions: none Substitute the current input and multiply by the coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore g(x) value := sum(1, -5) After g(x) value = -4 Conditions: none Then add this function's own constant term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore f(g(x)) product := mul(1, -4) After f(g(x)) product = -4 Conditions: none Substitute the current input and multiply by the coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore f(g(x)) value := sum(-4, -1) After f(g(x)) value = -5 Conditions: none Then add this function's own constant term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore g(f(-1))=-3;f(g(-1))=-5 After equal at supplied input=No Conditions: none Compare the supplied input only;one-point equality of arbitrary functions does not prove commutation everywhere.A separate affine coefficient identity could address this particular model globally. Step identifiers and dependenciesRule: pointwise_composition_comparison · Uses: G001, S009 ## Reverse checks - 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 f(g(x)) value with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute f(g(x)) product with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute g(x) value with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute g(x) product with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute g(f(x)) value with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute g(f(x)) product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute f(x) value with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute f(x) product 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 No 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: e89cc695505a97aac2ce756be3f60f5a1bda9f2f57d9c981861af6f7da8b28e4 - candidate record SHA-256: 2901049758c98d55f5028396beb98bc03573b29a9bb82dfea7a8c3dae7201323 - semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2 - question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260930-r04-composition-order-comparison-0012.json)