Definition of the inverse function

Bijectivity is exactly the condition that allows a function to be “inverted”.

Definition — Inverse function

If $f:A\to B$ is bijective, then for every $y\in B$ there exists exactly one $x\in A$ with $f(x)=y$. The inverse function $f^{-1}:B\to A$ is the one that associates with each $y\in B$ that $x$: $f^{-1}(y) = x \iff f(x) = y.$

Without bijectivity the inverse would not exist as a function: if $f$ were not injective, several $x$ would correspond to the same $y$ (violating uniqueness); if it were not surjective, for some $y$ no $x$ would exist.

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Topics: Functions and properties
Concepts: Bijective function · Inverse function