Two functions *f* and *g*
are **inverse functions** of each other if:

*f*(*g*(*x*)) =*x*whenever*x*∈ dom*g*, and*g*(*f*(*x*)) =*x*whenever*x*∈ dom*f*.

**Warning:**
*f*^{−1} does *not* mean 1/*f*!

There are two ways to caclulate *f*^{−1}:

- Start with
*x*=*f*(*y*) and solve for*y*; this gives you*y*=*f*^{−1}(*x*). - Start with
*y*=*f*(*x*) and solve for*x*; this gives you*x*=*f*^{−1}(*y*).

We have:

*f*^{−1}(*f*(*x*)) =*x*whenever*x*∈ dom*f*, and*f*(*f*^{−1}(*y*)) =*y*whenever*y*∈ ran*f*.

The graphs of inverse functions are related by switching *x* and *y*.
In particular:

- The range of
*f*is the same as the domain of*f*^{−1}; - the range of
*f*^{−1}is the same as the domain of*f*.

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2016 April 25. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`https://tobybartels.name/MATH-1150/2024FA/inverses/`

.