Chapter 4, Problem 23RE

In Problems $15-20$, graph each rational function following the seven steps on page $211$.

$R\left(x\right)=\frac{2x-6}{x}$

To determine

To graph: The rational function $R\left(x\right)=\frac{2x-6}{x}$.

Explanation of Solution

Given Information:

The rational function $R\left(x\right)=\frac{2x-6}{x}$.

Graph:

Step-1: The rational function $R\left(x\right)=\frac{2x-6}{x}$ becomes undefined when $x=0$. The domain of function $R\left(x\right)=\frac{2x-6}{x}$ is set of all real numbers except 0.

Therefore, domain of the rational function $R\left(x\right)=\frac{2x-6}{x}$ is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$.

Step-2: Since there is no common factor in numerator and denominator of rational function $R\left(x\right)=\frac{2x-6}{x}$. Function $R\left(x\right)$ is in lowest terms.

Step-3: Because $x\ne 0$, there is no $y-$ intercept.

If $R\left(x\right)=\frac{2x-6}{x}=0$, therefore, $2x-6=0$.

$\Rightarrow x=3$.

Therefore, function $R\left(x\right)=\frac{2x-6}{x}$ has $x-$ intercept $\left(3,0\right)$.

Step-4: Real zeros of denominator of function $R\left(x\right)=\frac{2x-6}{x}$ is $x=0$, so the graph of $R\left(x\right)$ has one vertical asymptote $x=0$ line.

Step-5: The degree of numerator is 1 and degree of denominator is 1. Since degree of denominator isequal to the degree of numerator, the graph of $R\left(x\right)$ has horizontalasymptote equals the ratio of the leading coefficient.

$\Rightarrow y=\frac{a_n}{b_m}=\frac{2}{1}=2$.

The line $y=2$ is a horizontal asymptote.

To find the graph of $R\left(x\right)=\frac{2x-6}{x}$ intersects horizontal asymptote $y=2$,

Solve equation $R\left(x\right)=2$.

$\Rightarrow \frac{2x-6}{x}=2$

$\Rightarrow 2x-6=2x$

$\Rightarrow -6=0$ which is impossible.

Therefore graph $R\left(x\right)$ does not cross horizontal asymptote $y=2$.

Step-6: The real zero of the numerator is $x=3$; the real zero of the denominator are $x=0$.

The intervals using the numbers $x=0,3$ are $\left(-\infty ,0\right),\left(0,3\right),\left(3,\infty \right)$.

$\begin{array}{cccc}\text{Intervals}& \left(-\infty ,0\right)& \left(0,3\right)& \left(3,\infty \right)\\ \text{Test points}& -3& 1& 4\\ \text{Value of R}\left(x\right)& 4& -4& \frac{1}{2}\\ \text{Location of graph}& \text{Above}\text{x-}\text{axis}& \text{Below}\text{x-}\text{axis}& \text{Above}\text{x-}\text{axis}\\ \text{Point on graph}& \left(-3,4\right)& \left(1,-4\right)& \left(4,\frac{1}{2}\right)\end{array}$

Step-7:

The graph of $R\left(x\right)$ is above $x-$ axis for $-\infty <x<0$ and graph e graph of $R\left(x\right)$ does not intersect $y=2$. Therefore, graph of $R\left(x\right)$ approaches $y=2$ from above as $x\to -\infty$.

Function $R\left(x\right)$ approaches to $\infty$ from left side of vertical asymptote $x=0$.

The graph of $R\left(x\right)$ is below $x-$ axis for $0<x<3$, the graph of $R\left(x\right)$ approaches to $-\infty$ as $R\left(x\right)$ approaches to vertical asymptote $x=0$ from right side.

Therefore, the graph of rational function $R\left(x\right)=\frac{2x-6}{x}$ is

Interpretation:

The graph of rational function $R\left(x\right)=\frac{2x-6}{x}$ is as shown above.