📂Functions

Inverse Function

Definition

For a given surjective function $f: X \to Y$, the inverse function of $f$ is defined as follows.

$$f^{-1} : Y \to X, \quad f^{-1}(y) = x \iff f(x) = y$$

A function for which an inverse function exists is called an invertible function.

Explanation

By definition, $f$ is the inverse function of $f^{-1}$.

$$f = (f^{-1})^{-1}$$

$f \circ f^{-1}$ and $f^{-1} \circ f$ are the identity functions on $Y$ and $X$, respectively $I_{Y}$, $I_{X}$.

$$f \circ f^{-1} : Y \to Y, \quad f \circ f^{-1}(y) = y \quad \forall y \in Y$$

$$f^{-1} \circ f : X \to X, \quad f^{-1} \circ f(x) = x \quad \forall x \in X$$