# Power Functions

A **power function** is any function of the form *f*(*x*) = *x*^{a}, where *a* is any real number.

### Sample Problem

The following are all power functions:

### Sample Problem

The following are all power functions, written deceptively.The function

is a power function since it can be written as

*f*(*x*) = *x*^{1/2}

or

*f*(*x*) = *x*^{.5}.

The function

is also a power function, since this can be written as

*g*(*x*) = *x ^{–}*

^{6}.

### Sample Problem

The function

*f*(*x*) = *x ^{x}*

is not a power function, because the exponent is a variable instead of a constant.

We usually assume the exponent *a* isn't 0, because if *a* is 0 we find a power function. The function

*f*(*x*) = *x*^{0} = 1

is a constant function, and we already know how to deal with those.

There's a skill needed for integrals that we'll consider now: thinking backwards. Instead of taking a function and figuring out its derivative, think about looking at a derivative and figuring out what sort of function it came from. Try this out when looking over solutions to derivatives.