Cube Root Parent Functions Solutions

Non-Polynomial Parent Functions
Type	Absolute Value	Square Root	Cube Root	Reciprocal
Parent equation	$y=\left|x\right|$	$y=\sqrt{x}$	$y=\sqrt[3]{x}$
Degree	does not apply	$\frac{1}{2}$	$\frac{1}{3}$
Domain and Range	$$\begin{gathered} \mathrm{domain}:\left\{x|x\in \mathrm{ℝ}\right\} \\ \mathrm{range}:\left\{y|y\in \mathrm{ℝ}, y\ge 0\right\} \end{gathered}$$	$$\begin{gathered} \mathrm{domain}:\left\{x|x\in \mathrm{ℝ}, x\ge 0\right\} \\ \mathrm{range}:\left\{y|y\in \mathrm{ℝ}, y\ge 0\right\} \end{gathered}$$	$$\begin{gathered} \mathrm{domain}:\left\{x|x\in \mathrm{ℝ}\right\} \\ \mathrm{range}:\left\{y|y\in \mathrm{ℝ}\right\} \end{gathered}$$
Quadrants	Q1 and Q2	Q1	Q1 and Q3
Because	y-coordinates will always be positive because $\left|\mathrm{term}\right|=+\mathrm{term}$	only $\mathrm{ℝ}$ are being graphed, so the x- and y-values will be zero	cube root of a number can be ±, so there are no restrictions for domain and range
Other Characteristics	v-shaped, symmetric Axis of symmetry is $x=0,$ the y-axis	partial inverse of the quadratic function	inverse of the cubic parent function
Intersects graph	vertex at origin, changes directions at origin	defined start and end point at the origin	crosses the coordinate plane at the origin

Example 8

Graph the cube root parent function. Name the domain and range in interval and set-builder notation. Then name the end behavior of the graph.

End behavior

As $x\to +\infty , f\left(x\right)\to +\infty ,$ and as $x\to –\infty , f\left(x\right)\to –\infty$