Identify power functions
[latex]A \left(r\right)=\pi {r}^{2}\[/latex]
and the function for the volume of a sphere with radius [latex]r[/latex] is
[latex]V \left(r\right)=\frac{4}{3}\pi {r}^{3}\[/latex]
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[latex]f\left(x\right)=k{x}^{p}[/latex]
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]