Chapter 2.7, Problem 23E

To determine

To sketch the function $f(x)=\frac{x^2}{x^2-4}$ , also find their intercepts, asymptotes and holes.

Answer to Problem 23E

  $x=\pm 2,y=1$

Explanation of Solution

Given:

Function: $f(x)=\frac{x^2}{x^2-4}$

Calculation for graph:

Consider $f(x)=\frac{x^2}{x^2-4}$

Values of x	Values of f (x)
0	0
1	-0.333
-1	-0.333
2	Infinity
-2	Infinity

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

Graph:

  

Calculation:

Intercepts:

Let $f(x)=y.$

  $\Rightarrow y=\frac{x^2}{x^2-4}$

To find x intercepts, put y = 0,

  $\begin{array}{l}\Rightarrow y=0\\ \Rightarrow \frac{x^2}{x^2-4}=0\\ \Rightarrow x^2=0\\ \Rightarrow x=0\end{array}$

So, x intercept is (0, 0)

To find y intercepts, put x = 0,

  $\begin{array}{l}\Rightarrow y=\frac{0}{0-4}\\ \Rightarrow y=\frac{0}{-4}\\ \Rightarrow y=0\end{array}$

So, y intercept is (0, 0).

Asymptotes:

Vertical asymptotes:

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

  $\begin{array}{l}\Rightarrow x^2-4=0\\ \Rightarrow x^2=4\\ \Rightarrow x=\pm \sqrt{4}\\ \Rightarrow x=\pm 2\end{array}$

Horizontal asymptotes:

As the degree of numerator is equal to the degree of the denominator, the horizontal asymptote is

  $\begin{array}{l}\Rightarrow y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\\ \Rightarrow y=\frac{1}{1}\\ \Rightarrow y=\text{1}\end{array}$

Holes:

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