Chapter 3.7, Problem 41E

To determine

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

Answer to Problem 41E

The rational function has $x$ -intercept is $x=1$ , $y$ -intercept is $y=-2$ , vertical asymptote is $x=-2$ , horizontal asymptote is $y=4$ , domain is $\left\{x|x\ne -2\right\}$ and range is $\left\{y|y\ne 4\right\}$ .

Explanation of Solution

Given information :

The rational function is $r\left(x\right)=\frac{4x-4}{x+2}$

Calculation:

Intercept:

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

  $\begin{array}{l}4x-4=0\\ 4x=4\\ x=1\end{array}$

So, $x$ -intercept is $x=1$ .

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

  $\begin{array}{l}r\left(0\right)=\frac{4\left(0\right)-4}{\left(0\right)+2}\\ r\left(0\right)=\frac{-4}{2}\\ r\left(0\right)=-2\end{array}$

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

Vertical asymptote:

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

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

The vertical asymptote is a line $x=-2$ .

Horizontal asymptote:

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

Since, $n=m$ ,

Therefore,

  $\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}=\frac{4}{1}=4$

So, the horizontal asymptote is the line $y=4$ .

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

$x$	$r\left(x\right)$
$-15$	4.9
$-10$	5.5
$-5$	8.0
$5$	2.3
$10$	3.0
$15$	3.3

The graph is obtained as:

  

From the above graph, it can be observed that the domain of the rational function is $\left\{x|x\ne -2\right\}$ and range is $\left\{y|y\ne 4\right\}$ .

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

  

Hence,

The rational function has $x$ -intercept is $x=1$ , $y$ -intercept is $y=-2$ , vertical asymptote is $x=-2$ , horizontal asymptote is $y=4$ , domain is $\left\{x|x\ne -2\right\}$ and range is $\left\{y|y\ne 4\right\}$ .