Power Functions

Before we can understand the bird problem, it will be helpful to understand a different 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.

As an example, consider functions for area or volume. The function for the area of a circle with radius \(r\) is

\(A(r)=\pi r^{2}\)

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

\(V(r)=\frac{4}{3} \pi r^{3}\)

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

POWER FUNCTION

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

\(f(x)=k x^{p}\)

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. 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?

\(\begin{array}{ll}

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)=\frac{1}{x} & \text { Reciprocal function } \\

f(x)=\frac{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{array}\)

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 \(f(x)=x^{\frac{1}{2}}\) or \(f(x)=x^{\frac{1}{3}}\).

TRY IT #1

Which functions are power functions?

\(\begin{aligned}

&f(x)=2 x \cdot 4 x^{3} \\

&g(x)=-x^{5}+5 x^{3} \\

&h(x)=\frac{2 x^{5}-1}{3 x^{2}+4}

\end{aligned}\)