Chapter 2.7, Problem 50E

To determine

To sketch the function $f(x)=\frac{x^3}{x^2-1}$ , also find their intercepts and asymptotes.

Answer to Problem 50E

  $x=1,-1,y=x$

Explanation of Solution

Given:

Function: $f(x)=\frac{x^3}{x^2-1}$

Calculation for graph:

Consider $f(x)=\frac{x^3}{x^2-1}$

Values of x	Values of f (x)
0	0
1	Infinity
-1	Infinity
2	2.666
-2	-2.666

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

Graph:

  

Calculation:

Intercepts:

Let $f(x)=y.$

  $\Rightarrow y=\frac{x^3}{x^2-1}$

To find x intercepts, put y = 0,

  $\begin{array}{l}\Rightarrow y=0\\ \Rightarrow \frac{x^3}{x^2-1}=0\\ \Rightarrow x^3=0\\ \Rightarrow x=0\end{array}$

So, the x intercept is (0, 0).

To find y intercepts, put x = 0,

  $\begin{array}{l}\Rightarrow y=\frac{{(0)}^3}{{(0)}^2-1}\\ \Rightarrow y=\frac{0}{-1}\\ \Rightarrow y=0\end{array}$

So, the y intercept is (0, 0).

Asymptotes:

Vertical asymptotes:

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

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

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: $x^3$

Divisor: $x^2-1$

  $x^2-1\begin{array}{c}\hfill x\\ \hfill \overline{)\begin{array}{l}{x}^3+0x^2+0x+0\\ \begin{array}{c}{x}^3+0x^2-x\end{array}\\ x\end{array}}\end{array}$  $\begin{array}{l}\frac{x^3}{x^2}=x\\ \text{Multiply}x(x^2-1)\text{and subtract}\text{.}\end{array}$

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

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