Chapter 3.7, Problem 43E

To determine

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

Answer to Problem 43E

The rational function has $x$ -intercept is $\frac{4}{3}$ , $y$ -intercept is $\frac{4}{7}$ , vertical asymptote is $x=-7$ , horizontal asymptote is $y=-3$ , domain is $\left\{x|x\ne -7\right\}$ and range is $\left\{y|y\ne -3\right\}$ .

Explanation of Solution

Given information :

The rational function is $r\left(x\right)=\frac{4-3x}{x+7}$

Calculation:

Intercept:

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

  $\begin{array}{l}4-3x=0\\ 3x=4\\ x=\frac{4}{3}\end{array}$

So, $x$ -intercept is $\frac{4}{3}$ .

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

  $\begin{array}{l}r\left(0\right)=\frac{4-3\left(0\right)}{\left(0\right)+7}\\ r\left(0\right)=\frac{4}{7}\end{array}$

The $y$ -intercept is $\frac{4}{7}$ .

Vertical asymptote:

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

  $\begin{array}{l}x+7=0\\ x=-7\end{array}$

The vertical asymptote is a line $x=-7$ .

Horizontal asymptote:

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

Since, $n=m$ ,

Therefore,

  $\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}=\frac{-3}{1}=-3$

So, the horizontal asymptote is the line $y=-3$ .

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

$x$	$r\left(x\right)$
$-18$	$-5.3$
$-12$	$-8$
$-6$	$22$
$6$	$-1.1$
$12$	$-1.7$
$18$	$-2$

The graph is obtained as:

  

From the above graph, it can be observed that the domain of the rational function is $\left\{x|x\ne -7\right\}$ and range is $\left\{y|y\ne -3\right\}$ .

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

  

Hence,

The rational function has $x$ -intercept is $\frac{4}{3}$ , $y$ -intercept is $\frac{4}{7}$ , vertical asymptote is $x=-7$ , horizontal asymptote is $y=-3$ , domain is $\left\{x|x\ne -7\right\}$ and range is $\left\{y|y\ne -3\right\}$ .