Identity Function

Identity Function

## Definition / Concept

The identity function maps every element of a set to itself.

## Formula

$$I_A:A\to A,\qquad I_A(x)=x$$

## Conditions / Special Cases

For any function:

$$f:A\to B$$

$$f\circ I_A=f$$

$$I_B\circ f=f$$

If $f$ is invertible:

$$f^{-1}\circ f=I_A$$

$$f\circ f^{-1}=I_B$$

## Usage

- Used to verify inverse functions.

- Acts as the identity element under function composition.