# Inverse function

https://arbital.com/p/inverse_function

by Michael Cohen Jun 24 2016 updated Jul 7 2016

The inverse of a function returns an input of the original function when fed the original's corresponding output.

If a function $g$ is the inverse of a function $f$, then $g$ undoes $f$, and $f$ undoes $g$. In other words, $g(f(x)) = x$ and $f(g(y)) = y$. An inverse function takes as its domain the range of the original function, and the range of the inverse function is the domain of the original. To put that another way, if $f$ maps $A$ onto $B$, then $g$ maps $B$ back onto $A$. To indicate the inverse of a function $f$, we write $f^{-1}$.

## Examples

$$y=f(x) = x^3\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = y^{1/3}$$ $$y=f(x) = e^x\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = ln(y)$$ $$y=f(x) = x+4\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = y-4$$

## Comments

Patrick Stevens

This page doesn't disambiguate between "left inverse" and "inverse". Strictly an "inverse" is a two-sided inverse, so gf = 1 and fg = 1.