Chapter 9, Problem 14STP

a.

To determine

To describe: the vertical and horizontal asymptote of the graph

Answer to Problem 14STP

The vertical asymptote is

Explanation of Solution

Given information: The graph provided in the question shown below

  

From the graph it is observed that the graph has two vertical asymptote and one horizontal asymptote

The vertical asymptotes are

  $\begin{array}{l}x=-4\\ x=4\end{array}$

The horizontal asymptote is

  $y=0$

b.

To determine

To write: the equation of the rational function

Answer to Problem 14STP

The equation of the rational function is $f\left(x\right)=\frac{b}{\left(x+4\right)\left(x-4\right)},b\in \left(0,16\right)$

Explanation of Solution

Given information: The graph provided in the question shown below

  

Calculation:

The vertical asymptote $x=-4\&x=4$ suggest the function is rational one containing denominator $x-4\&x+4$

The horizontal asymptote is $y=0$

  $\begin{array}{l}f\left(x\right)=\frac{ax+b}{\left(x+4\right)\left(x-4\right)}\\ f\left(x\right)=f\left(-x\right)\\ \frac{ax+b}{\left(x+4\right)\left(x-4\right)}=\frac{a\left(-x\right)+b}{\left(-x+4\right)\left(-x-4\right)}\\ ax+b=-ax+b\\ a=0\\ f\left(x\right)=\frac{b}{\left(x+4\right)\left(x-4\right)}\\ b>0\\ -1<f\left(0\right)<0\\ -1<\frac{b}{4\left(-4\right)}<0\\ -1<-\frac{b}{16}<0\\ 1>\frac{b}{16}\\ b<16\\ f\left(x\right)=\frac{b}{\left(x+4\right)\left(x-4\right)},b\in \left(0,16\right)\end{array}$