# 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.