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*.

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This web page was written between 2011 and 2016 by Toby Bartels, last edited on 2016 April 25. Toby reserves no legal rights to it.

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