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Study Guides > MTH 163, Precalculus

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 is

A(r)=πr2A\left(r\right)=\pi {r}^{2}\\

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

V(r)=43πr3V\left(r\right)=\frac{4}{3}\pi {r}^{3}\\

Both of these are examples of power functions because they consist of a coefficient, π\pi or 43π\frac{4}{3}\pi \\, 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

f(x)=kxpf\left(x\right)=k{x}^{p}\\

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

Q & A

Is f(x)=2xf\left(x\right)={2}^{x}\\ 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?

{f(x)=1Constant functionf(x)=xIdentify functionf(x)=x2Quadratic  functionf(x)=x3Cubic functionf(x)=1xReciprocal functionf(x)=1x2Reciprocal squared functionf(x)=xSquare root functionf(x)=x3Cube root function\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}\\

Solution

All of the listed functions are power functions.

The constant and identity functions are power functions because they can be written as f(x)=x0f\left(x\right)={x}^{0}\\ and f(x)=x1f\left(x\right)={x}^{1}\\ respectively.

The quadratic and cubic functions are power functions with whole number powers f(x)=x2f\left(x\right)={x}^{2}\\ and f(x)=x3f\left(x\right)={x}^{3}\\.

The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as f(x)=x1f\left(x\right)={x}^{-1}\\ and f(x)=x2f\left(x\right)={x}^{-2}\\.

The square and cube root functions are power functions with fractional powers because they can be written as f(x)=x1/2f\left(x\right)={x}^{1/2}\\ or f(x)=x1/3f\left(x\right)={x}^{1/3}\\.

Try It 1

Which functions are power functions?

{f(x)=2x24x3g(x)=x5+5x34xh(x)=2x513x2+4\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}\\

Solution

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