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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]Aleft(rright)=pi {r}^{2}\[/latex]

and the function for the volume of a sphere with radius r is

[latex]Vleft(rright)=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]fleft(xright)=k{x}^{p}\[/latex]

where k and p are real numbers, and k is known as the coefficient.

Q & A

Is [latex]fleft(xright)={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}fleft(xright)=1hfill & text{Constant function}hfill \ fleft(xright)=xhfill & text{Identify function}hfill \ fleft(xright)={x}^{2}hfill & text{Quadratic}text{ }text{ function}hfill \ fleft(xright)={x}^{3}hfill & text{Cubic function}hfill \ fleft(xright)=frac{1}{x} hfill & text{Reciprocal function}hfill \ fleft(xright)=frac{1}{{x}^{2}}hfill & text{Reciprocal squared function}hfill \ fleft(xright)=sqrt{x}hfill & text{Square root function}hfill \ fleft(xright)=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]fleft(xright)={x}^{0}\[/latex] and [latex]fleft(xright)={x}^{1}\[/latex] respectively.

The quadratic and cubic functions are power functions with whole number powers [latex]fleft(xright)={x}^{2}\[/latex] and [latex]fleft(xright)={x}^{3}\[/latex].

The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]fleft(xright)={x}^{-1}\[/latex] and [latex]fleft(xright)={x}^{-2}\[/latex].

The square and cube root functions are power functions with fractional powers because they can be written as [latex]fleft(xright)={x}^{1/2}\[/latex] or [latex]fleft(xright)={x}^{1/3}\[/latex].

Try It 1

Which functions are power functions?

[latex]begin{cases}fleft(xright)=2{x}^{2}cdot 4{x}^{3}hfill \ gleft(xright)=-{x}^{5}+5{x}^{3}-4xhfill \ hleft(xright)=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]..