5.2: Power Functions And Polynomial Functions

Identifying 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

\[A(r)={\pi}r^2 \nonumber\]

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

\[V(r)=\dfrac{4}{3}{\pi}r^3 \nonumber\]

Both of these are examples of power functions because they consist of a coefficient, \({\pi}\) or \(\dfrac{4}{3}{\pi}\), multiplied by a variable \(r\) raised to a power.

Definition: Power Function

A power function is a function that can be represented in the form

\[f(x)=kx^p \label{power}\]

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

Q&A: Is \(f(x)=2^x\) a power function?

No. A power function contains a variable base raised to a fixed power (Equation \ref{power}). This function has a constant base raised to a variable power. This is called an exponential function, not a power function. This function will be discussed later.

Example \(\PageIndex{1}\): Identifying Power Functions

Which of the following functions are power functions?

\[\begin{align*} f(x)&=1 &\text{Constant function} \\f(x)&=x &\text{Identify function} \\f(x)&=x^2 &\text{Quadratic function} \\ f(x)&=x^3 &\text{Cubic function} \\ f(x)&=\dfrac{1}{x} &\text{Reciprocal function} \\f(x)&=\dfrac{1}{x^2} &\text{Reciprocal squared function} \\ f(x)&=\sqrt{x} &\text{Square root function} \\ f(x)&=\sqrt[3]{x} &\text{Cube root function} \end{align*}\]

Solution

All of the listed functions are power functions.

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

The quadratic and cubic functions are power functions with whole number powers \(f(x)=x^2\) and \(f(x)=x^3\).

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

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

Exercise \(\PageIndex{1}\)

Which functions are power functions?

• \(f(x)=2x^2⋅4x^3\)
• \(g(x)=−x^5+5x^3−4x\)
• \(h(x)=\frac{2x^5−1}{3x^2+4}\)

Answer

\(f(x)\) is a power function because it can be written as \(f(x)=8x^5\). The other functions are not power functions.