A power function is any function of the form f(x) = xa, where a is any real number.
The following are all power functions:
The following are all power functions, written deceptively.The function
is a power function since it can be written as
f(x) = x1/2
f(x) = x.5.
is also a power function, since this can be written as
g(x) = x–6.
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.