Chapter 2.7, Problem 48E

To determine

To sketch the function $g(x)=\frac{1-x^2}{x}$ , also find their intercepts and asymptotes.

Answer to Problem 48E

  $x=0,y=-x$

Explanation of Solution

Given:

Function: $g(x)=\frac{1-x^2}{x}$

Calculation for graph:

Consider $g(x)=\frac{1-x^2}{x}$

Values of x	Values of g(x)
0	Undefined
1	0
-1	0
2	-1.5
-2	1.5

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

Graph:

  

Calculation:

Intercepts:

Let $g(x)=y.$

  $\Rightarrow y=\frac{1-x^2}{x}$

To find x intercepts, put y = 0,

  $\begin{array}{l}\Rightarrow y=0\\ \Rightarrow \frac{1-x^2}{x}=0\\ \Rightarrow 1-x^2=0\\ \Rightarrow x^2=1\\ \Rightarrow x=\pm \sqrt{1}\\ \Rightarrow x=\pm 1\end{array}$

So, the x intercepts are (1, 0) and (-1, 0).

To find y intercepts, put x = 0,

  $\begin{array}{l}\Rightarrow y=\frac{1-{(0)}^2}{0}\\ \Rightarrow y=\frac{1-0}{0}\\ \Rightarrow y=\frac{1}{0}\to \text{Undefined}\end{array}$

So, there are no y intercepts for given function.

Asymptotes:

Vertical asymptotes:

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

  $\Rightarrow x=0$

Horizontal asymptotes:

As the degree of numerator is larger than the degree of the denominator, the horizontal asymptote does not exist for given function.

Slant asymptotes:

To find slant asymptote, divide given function numerator with denominator using long division.

Dividend: $1-x^2$

Divisor: $x$

  $x\begin{array}{c}\hfill -x\\ \hfill \overline{)\begin{array}{l}1-x^2\\ \begin{array}{c}0-x^2\end{array}\\ 1\end{array}}\end{array}$  $\begin{array}{l}\frac{-x^2}{x}=-x\\ \text{Multiply}-x(x)\text{and subtract}\text{.}\end{array}$

Here, quotient is $-x$ and remainder is 1.

So, the slant asymptote is $y=-x.$