Reciprocal Parent Functions Solutions

Non-Polynomial Parent Functions
Type	Absolute Value	Square Root	Cube Root	Reciprocal
Parent equation	$y=\left|x\right|$	$y=\sqrt{x}$	$y=\sqrt[3]{x}$	$y=\frac{1}{x}, x\ne 0$
Degree	does not apply	$\frac{1}{2}$	$\frac{1}{3}$	–1
Domain and Range	$$\begin{gathered} \mathrm{domain}:\left\{x|x\in \mathrm{ℝ}\right\} \\ \mathrm{range}:\left\{y|y\in \mathrm{ℝ}, y\ge 0\right\} \end{gathered}$$	$$\begin{gathered} \mathrm{domain}:\left\{x|x\in \mathrm{ℝ}, x\ge 0\right\} \\ \mathrm{range}:\left\{y|y\in \mathrm{ℝ}, y\ge 0\right\} \end{gathered}$$	$$\begin{gathered} \text{domain}:\left\{x|x\in \mathrm{ℝ}\right\} \\ \mathrm{range}:\left\{y|y\in \mathrm{ℝ}\right\} \end{gathered}$$	$$\begin{gathered} \mathrm{domain}:\left\{x|x\in \mathrm{ℝ}, x\ne 0\right\} \\ \mathrm{range}:\left\{y|y\in \mathrm{ℝ}, y\ne 0\right\} \end{gathered}$$
Quadrants	Q1 and Q2	Q1	Q1 and Q3	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	1 divided by a positive value will give a positive value $(+x, +y)$; 1 divided by a negative value will give a negative value $(–x, –y)$
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	has asymptotes denominator ≠ zero
Intersects graph	vertex at origin, changes directions at origin	defined start and end point at the origin	crosses the coordinate plane at the origin	continues across the coordinate plane in four directions with four end behavior statements (not two)

Example 9

Graph the rational parent function. Name the domain and range set-builder notation. Then name the end behavior of the graph.

End behavior

Quadrant I
As $x\to +\infty , f\left(x\right)\to 0,$ and as $x\to 0^{+}, f\left(x\right)\to +\infty$

Quadrant III
As $x\to –\infty , f\left(x\right)\to 0,$ and as $x\to 0^{–}, f\left(x\right)\to –\infty$

The direction as x approaches zero is noted here because the rational parent function is in two quadrants.

Note

Find more details about graphing rational functions in Unit 1. Using interval notation for rational functions requires the use of union ∪ and intersection ⋂ symbols. This has not yet been introduced, therefore only set-builder notation is required at this time.

$\mathrm{domain}: \left(–\infty , 0\right)\cup \left(0, \infty \right),$$\mathrm{range}: \left(–\infty , 0\right)\cup \left(0, \infty \right)$