# Inverse function

From Calculus

An inverse function is a function that serves to "undo" another function. That is, if **f(x)** produces **y**, then putting **y** into the inverse of **f** produces the output **x**. Not every function has an inverse.

## Definition

A function **g** is the inverse function of f if **f(g(x))**=**x** for each value of **x** in the domain of **g**, and **g(f(x))=x** for each value of **x** in the domain of **f**. The function **g** is denoted as **f ^{-1}** ("inverse of f").

### Notation

Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, **f ^{-1}(x)**≠

**1/f(x)**.

## Relevant observations

- If
**g**is the inverse function of**f**, then**f**is the inverse function of**g**. - The domain of
**f**is the range of^{-1}**f**and the range of**f**is the domain of^{-1}**f**. - A function may not have an inverse function, but if it has, the inverse function is unique.