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Study Guides > College Algebra

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 r is

[latex]A\left(r\right)=\pi {r}^{2}\\[/latex]

and the function for the volume of a sphere with radius r 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 r 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)=1\hfill & \text{Constant function}\hfill \\ f\left(x\right)=x\hfill & \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]\begin{cases}f\left(x\right)=2{x}^{2}\cdot 4{x}^{3}\hfill \\ g\left(x\right)=-{x}^{5}+5{x}^{3}-4x\hfill \\ h\left(x\right)=\frac{2{x}^{5}-1}{3{x}^{2}+4}\hfill \end{cases}\\[/latex]

Solution

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  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..