Chapter 2.7, Problem 24E

To determine

To sketch the function $g(x)=\frac{x}{x^2-9}$ , also find their intercepts, asymptotes and holes.

Answer to Problem 24E

  $x=\pm 3,y=0$

Explanation of Solution

Given:

Function: $g(x)=\frac{x}{x^2-9}$

Calculation for graph:

Consider $g(x)=\frac{x}{x^2-9}$

Values of x	Values of g(x)
0	0
1	-0.125
-1	0.125
2	-0.4
-2	0.4

By taking different values of x, the graph can be plotted.

Graph:

  

Calculation:

Intercepts:

Let $g(x)=y.$

  $\Rightarrow y=\frac{x}{x^2-9}$

To find x intercepts, put y = 0,

  $\begin{array}{l}\Rightarrow y=0\\ \Rightarrow \frac{x}{x^2-9}=0\\ \Rightarrow x=0\end{array}$

So, x intercept is (0, 0)

To find y intercepts, put x = 0,

  $\begin{array}{l}\Rightarrow y=\frac{0}{0-9}\\ \Rightarrow y=\frac{0}{-9}\\ \Rightarrow y=0\end{array}$

So, y intercept is (0, 0).

Asymptotes:

Vertical asymptotes:

To find vertical asymptotes, put denominator of the given function equal to zero.

  $\begin{array}{l}\Rightarrow x^2-9=0\\ \Rightarrow x^2=9\\ \Rightarrow x=\pm \sqrt{9}\\ \Rightarrow x=\pm 3\end{array}$

Horizontal asymptotes:

As the degree of numerator is less than the degree of the denominator, the horizontal asymptote is $y=0.$

Holes:

Here, the given function does not contain common factor in numerator and denominator. So, the function does not have any holes.