and the function for the volume of a sphere with radius [latex]r[/latex] is
Both of these are examples of power functions because they consist of a coefficient, [latex]\pi [/latex] or [latex]\frac{4}{3}\pi [/latex], multiplied by a variable [latex]r[/latex] raised to a power.
A General Note: Power Function
A power function is a function that can be represented in the form
where k and p are real numbers, and k is known as the coefficient.
Q & A
Is [latex]f\left(x\right)={2}^{x}[/latex] 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?
[latex]begin{cases}f\left(x\right)=1hfill & text{Constant function}hfill \ f\left(x\right)=xhfill & text{Identify function}hfill \ f\left(x\right)={x}^{2}hfill & text{Quadratic}text{ }text{ function}hfill \ f\left(x\right)={x}^{3}hfill & text{Cubic function}hfill \ f\left(x\right)=\frac{1}{x} hfill & text{Reciprocal function}hfill \ f\left(x\right)=\frac{1}{{x}^{2}}hfill & text{Reciprocal squared function}hfill \ f\left(x\right)=sqrt{x}hfill & text{Square root function}hfill \ f\left(x\right)=sqrt[3]{x}hfill & text{Cube root function}hfill end{cases}[/latex]
Solution
All of the listed functions are power functions.
The constant and identity functions are power functions because they can be written as [latex]f\left(x\right)={x}^{0}[/latex] and [latex]f\left(x\right)={x}^{1}[/latex] respectively.
The quadratic and cubic functions are power functions with whole number powers [latex]f\left(x\right)={x}^{2}[/latex] and [latex]f\left(x\right)={x}^{3}[/latex].
The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f\left(x\right)={x}^{-1}[/latex] and [latex]f\left(x\right)={x}^{-2}[/latex].
The square and cube root functions are power functions with \fractional powers because they can be written as [latex]f\left(x\right)={x}^{1/2}[/latex] or [latex]f\left(x\right)={x}^{1/3}[/latex].
Try It 1
Which functions are power functions?
[latex]f\left(x\right)=2{x}^{2}\cdot4{x}^{3}[/latex]
[latex]g\left(x\right)=-{x}^{5}+5{x}^{3}-4x[/latex]
[latex]h\left(x\right)=\frac{2{x}^{5}-1}{3{x}^{2}+4}[/latex]
Solution