Inverse Function

Inverse Function

## Definition / Concept

If $$f$$ is bijective, its inverse $$f^{-1}$$ maps each output back to its unique corresponding input.

## Formula

$$y=f(x)\Rightarrow x=f^{-1}(y)$$

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

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

## Conditions / Special Cases

The inverse exists only when $f$ is bijective between its specified domain and codomain.

The domain and codomain interchange:

$$\text{Domain}(f^{-1})=\text{Codomain}(f)$$

$$\text{Codomain}(f^{-1})=\text{Domain}(f)$$

## Usage

- Used to find inverse functions and solve equations involving compositions.