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
and the function for the volume of a sphere with radius r is
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
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]
SolutionLicenses & Attributions
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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]..
