# Inverse functions of bijections

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

For a bijection f:A→B, an inverse g:B→A restores values in both directions; exchanging a table is not an unconditional inverse construction.

mko-inverse-function · definition · v0.10.0



## Mathematical statement

If f:A→B is bijective, its inverse g:B→A assigns each b its unique preimage and satisfies g(f(x))=x for x∈A and f(g(y))=y for y∈B. A finite bijection table can be inverted by exchanging ordered pairs. For nonbijective f, the exchanged table is not automatically an inverse on all of B.





## Representative formula (exact TeX)

g(f(x))=x





## Assumptions


- Inverse-table and evaluation exercises first establish that f is a bijection on the declared A→B.
- A candidate g is first validated as a total function B→A, then both identities are checked. The representative formula shows only one side.
- Affine examples use f:R→R with m≠0; inverse coefficients are given as reduced rational numbers with positive denominators.





## Symbol scope


- g — inverse or candidate total function
- f — declared bijection
- x — permitted input in A





## Intuition, definition, and misconceptions

Every target occurs exactly once in a finite bijection table, so exchanging pairs produces a total function B→A. The inverse of the empty bijection ∅→∅ is the empty function.

An inverse satisfies both identities. Candidate checks in this pulse verify each side and declare both types completely, without identifying a general one-sided inverse with a two-sided inverse. An inverse function is distinct from the reciprocal 1/f(x).

For f(x)=mx+b with m≠0, g(y)=(y−b)/m. In the form g(y)=uy+v, u=1/m and v=−b/m. Exercises compute exact rational coefficients from integer parameters; a machine proof of general invertibility is not claimed.

Common errors include calling an exchanged nonbijective table an inverse function, or interpreting f⁻¹ as 1/f.





## Adopted derivation guide

[Inverse functions: exchanging a finite bijection table and inverting an affine map with nonzero slope](https://ocme.evemisslab.com/en/derivations/derivation-mko-inverse-function-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/mko/mko-bijective/) — An inverse requires a unique preimage for every target.
- [Function composition](https://ocme.evemisslab.com/en/mko/mko-composition/) — Composition expresses restoration in both directions.
- [Affine functions and degree-one functions](https://ocme.evemisslab.com/en/mko/mko-linear-function/) — The real affine example has nonzero slope.





## Verification boundary

Explicit finite bijections allow verification of both restoration identities; general inverse existence, uniqueness, and type bridges await formalization.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bind Function.LeftInverse, Function.RightInverse, Equiv.symm, and inverses of affine functions with nonzero slope.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/inverse-function.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-inverse-function/index.md)