Chapter 3.4, Problem 38E

To determine

To graph: the equation using parent graph $p\left(x\right)$ .

Explanation of Solution

Given information:

  $y=2\sqrt[5]{x-4}$

  $p\left(x\right)=x^5$

Calculation:

The $p^{-1}\left(x\right)$ can be evaluated as follows:

  $\begin{array}{c}y=x^5\left[\text{Let }p\left(x\right)=y\right]\\ x=y^5\left[\text{Interchange }x\text{ and }y\right]\\ y=\sqrt[5]{x}\left[\text{Solve for }y\right]\\ p^{-1}\left(x\right)=\sqrt[5]{x}\left[\text{Replace }y\text{ with }{p}^{-1}\left(x\right)\right]\end{array}$

Now make a table of values for $p^{-1}\left(x\right)$ as shown below.

$p\left(x\right)$	$x$	$-2$	$-1$	0	1	2	3	4
	$y$	$-32$	$-1$	0	1	32	243	1024

$p^{-1}\left(x\right)$	$x$	$-2$	$-1$	0	1	2	4
	$y$	$-1.14869$	$-1$	0	1	1.14869	$1.31950$

The graph of $p\left(x\right)\text{ and }{p}^{-1}\left(x\right)$ is shown below.

  

The red line represents $p\left(x\right)$ and blue line represents $p^{-1}\left(x\right)$ .

Shift the $p^{-1}\left(x\right)$ graph to right by 4 units and stretch it by a factor of 2 to get the resulting graph $y$ (denoted by green) as follows: