{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-inverse-function"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-inverse-function-01","title":{"zh-Hant":"反函數：交換有限雙射表並求非零斜率仿射函數的反函數","en":"Inverse functions: exchanging a finite bijection table and inverting an affine map with nonzero slope"},"summary":{"zh-Hant":"（一）A={1,2,3}、B={4,5,6}，p(1)=5、p(2)=6、p(3)=4。先確認 p 是雙射，再交換有序對得到候選 q，驗證 q 是總函數 B→A，並逐側檢查 q(p(x))=x 與 p(q(y))=y。（二）f:ℝ→ℝ，f(x)=−4x+6（m=−4≠0）：求反函數 g(y)=uy+v，u、v 以最簡、正分母的有理數表示，並對所有實數驗證兩側恆等式。（三）說明 h(1)=4、h(2)=4、h(3)=6 的交換表不是 B→A 的函數，以及 f^{-1} 不是 1/f。","en":"(1) A={1,2,3}, B={4,5,6}, p(1)=5, p(2)=6, p(3)=4. First confirm that p is a bijection, then exchange its ordered pairs to get a candidate q, validate q as a total function B→A, and check q(p(x))=x and p(q(y))=y separately. (2) f:ℝ→ℝ, f(x)=−4x+6 (m=−4≠0): find the inverse g(y)=uy+v with u, v reduced rationals with positive denominators, and verify both identities for all reals. (3) Show that the exchanged table of h(1)=4, h(2)=4, h(3)=6 is not a function B→A, and that f^{-1} is not 1/f."},"mko_ids":["mko-inverse-function"]}],"title":"Concept derivation guides"}