Power Functions

Power Function

A power function is a function that can be represented in the form

$f(x)=k x^{p}$

where $k$ and $p$ are real numbers, and $k$ is known as the coefficient.

Q&A

Is $f(x)=2^{x}$ a power function?

No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.

Example 1

Identifying Power Functions

Which of the following functions are power functions?

$\begin{array}{ll} f(x)=1 & \text { Constant function } \\ f(x)=x & \text { Identify function } \\ f(x)=x^{2} & \text { Quadratic function } \\ f(x)=x^{3} & \text { Cubic function } \\ f(x)=\frac{1}{x} & \text { Reciprocal function } \\ f(x)=\frac{1}{x^{2}} & \text { Reciprocal squared function } \\ f(x)=\sqrt{x} & \text { Square root function } \\ f(x)=\sqrt[3]{x} & \text { Cube root function } \end{array}$

Solution

All of the listed functions are power functions.

The constant and identity functions are power functions because they can be written as $f(x)=x^{0}$ and $f(x)=x^{1}$ respectively.

The quadratic and cubic functions are power functions with whole number powers $f(x)=x^{2}$ and $f(x)=x^{3}$.

The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as $f(x)=x^{-1}$ and $f(x)=x^{-2}$.

The square and cube root functions are power functions with fractional powers because they can be written as $f(x)=x^{\frac{1}{2}}$ or $f(x)=x^{\frac{1}{3}}$.

Try It #1

Which functions are power functions?

$\begin{aligned} &f(x)=2 x \cdot 4 x^{3} \\ &g(x)=-x^{5}+5 x^{3} \\ &h(x)=\frac{2 x^{5}-1}{3 x^{2}+4} \end{aligned}$