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A power function is any function of the form f(x) = xa, 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) = x1/2

or

f(x) = x.5.

The function

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

g(x) = x6.

Sample Problem

The function 

f(x) = xx

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) = x0 = 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.

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