Chapter 2.7, Problem 54E

To determine

To sketch the function $f(x)=\frac{x^3}{x^2+4}$ , also find their intercepts and asymptotes.

Answer to Problem 54E

  $y=x$

Explanation of Solution

Given:

Function: $f(x)=\frac{x^3}{x^2+4}$

Calculation for graph:

Consider $f(x)=\frac{x^3}{x^2+4}$

Values of x	Values of f (x)
0	0
1	0.2
-1	-0.2
2	1
-2	-1

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

Graph:

  

Calculation:

Intercepts:

Let $f(x)=y.$

  $\Rightarrow y=\frac{x^3}{x^2+4}$

To find x intercepts, put y = 0,

  $\begin{array}{l}\Rightarrow y=0\\ \Rightarrow \frac{x^3}{x^2+4}=0\\ \Rightarrow x^3=0\\ \Rightarrow x=0\end{array}$

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

To find y intercepts, put x = 0,

  $\begin{array}{l}\Rightarrow y=\frac{{(0)}^3}{{(0)}^2+4}\\ \Rightarrow y=\frac{0}{4}\\ \Rightarrow y=0\end{array}$

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

Asymptotes:

Vertical asymptotes:

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

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

So, vertical asymptote does not exist for given function.

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: $x^2+4$

  $x^2+4\begin{array}{c}\hfill x\\ \hfill \overline{)\begin{array}{l}{x}^3+0x^2+0x+0\\ \begin{array}{c}{x}^3+0x^2+4x\end{array}\\ -4x\end{array}}\end{array}$  $\begin{array}{l}\frac{x^3}{x^2}=x\\ \text{Multiply}x(x^2+4)\text{and subtract}\text{.}\end{array}$

Here, quotient is $x$ and remainder is $-4x.$

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