Chapter 3.7, Problem 46E

To determine

To find the intercept, asymptotes, domain, range and sketch the graph of the rational function.

Answer to Problem 46E

The rational function has no $x$ -intercept $-2$ , $y$ -intercept is $2$ , vertical asymptote is $x=-1$ , horizontal asymptote is $y=0$ , domain is $\left\{x|x\ne -1\right\}$ and range is $\left\{y|y<0\right\}$ .

Explanation of Solution

Given information :

The rational function is $r\left(x\right)=\frac{x-2}{{\left(x+1\right)}^2}$

Calculation:

Intercept:

The $x$ -intercept are the zeros of the numerator,

  $\begin{array}{l}x-2=0\\ x=2\end{array}$

So, $x$ -intercept is $2$ .

To find $y$ -intercept, substitute $x=0$ into original form of the function,

  $\begin{array}{l}r\left(0\right)=\frac{\left(0\right)-2}{{\left(0+1\right)}^2}\\ r\left(0\right)=\frac{-2}{1}\\ r\left(0\right)=-2\end{array}$

The $y$ -intercept is $-2$ .

Vertical asymptote:

The vertical asymptote occurs where the denominator is 0, that is, where the function is undefined.

  $\begin{array}{l}x+1=0\\ x=-1\end{array}$

The vertical asymptote is a line $x=-1$ .

Horizontal asymptote:

Here, the degree of the numerator is $n=1$ and denominator is $m=2$ .

Since, $n<m$ ,

So, the horizontal asymptote is the line $y=0$ .

Use the above information together with some additional values which is show in table below to sketch the graph:

$x$	$r\left(x\right)$
$-8$	$-0.2$
$-6$	$-0.3$
$-4$	$-0.7$
$1$	$-0.3$
$3$	$0.1$
$8$	$0.1$

The graph is obtained as:

  

From the above graph, it can be observed that the domain of the rational function is $\left\{x|x\ne -1\right\}$ and range is $\left\{y|y<0\right\}$ .

To check the result graph the rational function using graphing calculator,

  

Hence,

The rational function has no $x$ -intercept $-2$ , $y$ -intercept is $2$ , vertical asymptote is $x=-1$ , horizontal asymptote is $y=0$ , domain is $\left\{x|x\ne -1\right\}$ and range is $\left\{y|y<0\right\}$ .