Chapter 3.7, Problem 50E

To determine

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

Answer to Problem 50E

The rational function has $x$ -intercept is 2, $y$ -intercept is $2$ , vertical asymptote is $x=1\text{ and }x=-2$ , horizontal asymptote is $y=0$ , domain is $\left\{x|x\ne 1,-2\right\}$ and range is $ℝ$ .

Explanation of Solution

Given information :

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

Calculation:

Intercept:

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

  $\begin{array}{l}2x-4=0\\ 2x=4\\ x=2\end{array}$

So, $x$ -intercept is 2.

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

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

The $y$ -intercept is $2$ .

Vertical asymptote:

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

  $x^2+x-2=0$

Use Quadratic formula to solve the above equation,

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Here, $a=1,b=1\text{ and }c=-2$

  $\begin{array}{l}x=\frac{-\left(1\right)\pm \sqrt{{\left(1\right)}^2-4\left(1\right)\left(-2\right)}}{2\left(1\right)}\\ x=\frac{-1\pm \sqrt{1+8}}{2}\\ x=\frac{-1\pm \sqrt{9}}{2}\\ x=\frac{-1\pm 3}{2}\end{array}$

Either,

  $\begin{array}{l}x=\frac{-1+3}{2}\\ x=\frac{2}{2}\\ x=1\end{array}$

Or,

  $\begin{array}{l}x=\frac{-1-3}{2}\\ x=\frac{-4}{2}\\ x=-2\end{array}$

The vertical asymptote is a line $x=1$ and $x=-2$ .

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 (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 $\left\{x|x\ne 1,-2\right\}$ and range is $ℝ$ .

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

  

Hence,

The rational function has $x$ -intercept is 2, $y$ -intercept is $2$ , vertical asymptote is $x=1\text{ and }x=-2$ , horizontal asymptote is $y=0$ , domain is $\left\{x|x\ne 1,-2\right\}$ and range is $ℝ$ .