# Inverse function

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

$$y=f(x) = x^3\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = y^{1/3}$$

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

$$y=f(x) = x+4\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = y-4$$

Parents:

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

Good point!