# What Are Power and Root Functions?

A power function is a special case of a polynomial function. A power function consists of only one term—a polynomial of the form $a{x}^{n}$.

Theory

### PowerandRootFunctions

A power function is a function where $f\left(x\right)$ is given as a number multiplied by an arbitrary power of $x$. The function can be written as

 $f\left(x\right)=a\cdot {x}^{n}$

If $n$ is a fraction, we call the power function a root function, since it can be rewritten using the formula

 ${x}^{\frac{m}{n}}=\sqrt[n]{{x}^{m}}$

Below is a brief description of how each function behaves for different values of $n$.

• If $n$ is a even number, then you get a parabola.

• If $n$ is an odd number, you get graphs that are extended along the entire $y$-axis.

• If $n=0$, you get a straight line that intersects $y=a$.

• If $n<0$, you get rational functions.

• If $n\in ℚ$ ($n$ is a fraction), you get a root function.

• If $n$ is in the form $n=\frac{k}{2m}$, and if $k$ and $2m$ have no factors in common, then the graph begins in the origin.

Note! Root functions are defined only for positive values of $x$, since you can only take the even root ($\sqrt{x}$, $\sqrt[4]{x}$, $\sqrt[6]{x}$,$\dots$) of numbers greater than or equal to 0.

Example 1

 $f\left(x\right)=5{x}^{\frac{1}{2}}=5\sqrt{x}$

is a power function and a root equation

Example 2

 $f\left(x\right)=2{x}^{4}$

is a power function and a polynomial function.

Example 3

 $f\left(x\right)=4{x}^{-2}=\frac{4}{{x}^{2}}$

is a power function and a rational function.