Chapter 8.4, Problem 10PPS

To determine

To graph: The given function.

Explanation of Solution

Given information:

The following function:

  $f\left(x\right)=\frac{x^4-16}{x^2-1}$

Calculation:

Step 1:

Find the zeroes.

  $\begin{array}{l}{x}^4-16=0[\text{Set }a(x)=0]\\ \left(x^2+4\right)\left(x^2-4\right)=0[\text{Factor]}\\ x^2+4=0x^2-4=0[\text{Set each factor to 0]}\\ x^2=-4x^2=4[\text{Simplify]}\\ x=\sqrt{-4}x=\sqrt{4}[\text{Take square root of both sides]}\\ \text{No real }x=\pm 2[\text{Find square root]}\\ \text{solution}\text{.}\end{array}$

There is a zero at $x=\pm 2$ .

Step 2:

Find the asymptotes.

  $\begin{array}{l}{x}^2-1=0[\text{Set }b(x)=0]\\ \left(x+1\right)\left(x-1\right)=0[\text{Factor]}\\ x+1=0x-1=0[\text{Set each factor to 0]}\\ x=-1x=1[\text{Simplify]}\end{array}$

There is a vertical asymptote at $x=-1$ and $x=1$ .

The degree of numerator is greater than the degree of the denominator.

So, there is no horizontal asymptote.

Step 3:

Draw the graph

Use a table to find ordered pairs on the graph. Then connect the points