Identify power functions

In order to better understand the bird problem, we need to understand a specific type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.) As an example, consider functions for area or volume. The function for the area of a circle with radius [latex]r[/latex] is

[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


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]


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?