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

Power and Root Functions

A power function is a function where

f (x)

is given as a number multiplied by an arbitrary power of

x

. The function can be written as

f (x) = 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.

$Graph of a power function where n is a fraction$
If $n$ is in the form $n = \frac{k}{2 m}$ , and if $k$ and $2 m$ have no factors in common, then the graph begins in the origin.

$Graph of a power function where n is a reduced fraction$

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 (x) = 5 x^{\frac{1}{2}} = 5 \sqrt{x}

is a power function and a root equation

Example 2

f (x) = 2 x^{4}

is a power function and a polynomial function.

Example 3

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

is a power function and a rational function.

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