# Identify power functions

**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

[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]

### 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-display]f\left(x\right)=2{x}^{2}\cdot4{x}^{3}[/latex-display] [latex-display]g\left(x\right)=-{x}^{5}+5{x}^{3}-4x[/latex-display] [latex-display]h\left(x\right)=\frac{2{x}^{5}-1}{3{x}^{2}+4}[/latex-display] Solution## Licenses & Attributions

### CC licensed content, Shared previously

- Precalculus.
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