Chapter 3.7, Problem 49E

To determine

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

Answer to Problem 49E

The rational function has no $x$ -intercept, $y$ -intercept is $-1$ , vertical asymptote is $x=-1\text{ and }x=6$ , horizontal asymptote is $y=0$ , domain is $\left\{x|x\ne -1,6\right\}$ and range is $ℝ$ .

Explanation of Solution

Given information :

The rational function is $s\left(x\right)=\frac{6}{x^2-5x-6}$

Calculation:

Intercept:

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

Since, there is no real root of the numerator.

So, there is no $x$ -intercept.

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

  $\begin{array}{l}s\left(0\right)=\frac{6}{{\left(0\right)}^2-5\left(0\right)-6}\\ s\left(0\right)=\frac{6}{-6}\\ s\left(0\right)=-1\end{array}$

The $y$ -intercept is $-1$ .

Vertical asymptote:

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

  $x^2-5x-6=0$

Use Quadratic formula to solve the above equation,

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Here, $a=1,b=-5\text{ and }c=-6$

  $\begin{array}{l}x=\frac{-\left(-5\right)\pm \sqrt{{\left(-5\right)}^2-4\left(1\right)\left(-6\right)}}{2\left(1\right)}\\ x=\frac{5\pm \sqrt{25+24}}{2}\\ x=\frac{5\pm \sqrt{49}}{2}\\ x=\frac{5\pm 7}{2}\end{array}$

Either,

  $\begin{array}{l}x=\frac{5+7}{2}\\ x=\frac{12}{2}\\ x=6\end{array}$

Or,

  $\begin{array}{l}x=\frac{5-7}{2}\\ x=\frac{-2}{2}\\ x=-1\end{array}$

The vertical asymptote is a line $x=-1$ and $x=6$ .

Horizontal asymptote:

Here, the degree of the numerator is $n=0$ 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 (by choosing some test point) to sketch the graph.

The graph is obtained as:

  

From the above graph, it can be observed that the domain of the rational function is $\left\{x|x\ne -1,6\right\}$ and range is $ℝ$ .

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

  

Hence,

The rational function hasno $x$ -intercept, $y$ -intercept is $-1$ , vertical asymptote is $x=-1\text{ and }x=6$ , horizontal asymptote is $y=0$ , domain is $\left\{x|x\ne -1,6\right\}$ and range is $ℝ$ .