Question

Discuss each rational function following Steps 1–7 given on page 228.

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

Short answer

The domain is $\left\{x|x\ne 0\right\}$and the graph of the function is

Step-by-step solution

Step 1. Given Information

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

Step 2. Explanation

Factor the numerator and denominator of the given function.

$R\left(x\right)=\frac{2(x-3)}{x}$

The domain is $\left\{x|x\ne 0\right\}$

Since 0 is not in the domain, there is no y-intercept.

The real zero of the numerator satisfies the equation $x-3=0$

Hence, the only x-intercept is 3.

Step 3. Explanation  

Determine the behavior of the graph near the x-intercept by substituting 3 in the denominator.

$R\left(x\right)\approx \frac{2}{3}(x-3)$

Plot the point $(3,0)$and indicate a line with slope $\frac{2}{3}$on the graph.

The line $x=0$is the vertical asymptote of the graph.

The degree of numerator is equal to degree of denominator. So, the function is improper and graph has horizontal asymptote.

The leading coefficient of numerator and denominator is 2 and 1 respectively that is,

$$\begin{gathered} y=\frac{2}{1} \\ y=2 \end{gathered}$$

Step 4. Calculation

Substitute $R\left(x\right)=2$to determine whether the graph intersects the horizontal asymptote.

$$\begin{gathered} \frac{2(x-3)}{x}=2 \\ 2x-6=2x \\ -6\ne 0 \end{gathered}$$

There is no solution for x. So, the graph does not intersect the horizontal asymptote.

Construct the table and determine the values in the three intervals.

Step 5. Graph

Plot the points on the graph.