Chapter 4.3, Problem 38AYU

In Problems 7-50, follow Steps 1 through 7 on page 234 to analyze the graph of each function.

$R\left(x\right)=\frac{x^2+x-30}{x+6}$

To determine

To find: Analyze the given function.

Answer to Problem 38AYU

Explanation of Solution

Given:

$R\left(x\right)=\frac{x^2+x-30}{x+6}$

Step 1: Factor $R$ to get.

$R\left(x\right)=\frac{x^2+x-30}{x+6}$

$R\left(x\right)=\frac{\left(x+6\right)\left(x-5\right)}{x+6}$

That is $x=- 6$.

The domain $\left\{x\text{|}x\ne -6\right\}$.

Step 2: $R$ is in lowest terms $\left(x-5\right)$.

Step 3: Locate the intercepts of the graph. Use multiplicity to determine the bahaviour of the graph of $R$ at each $x\text{-intercept}$.

$y\text{-intercept}$ $- 5$.

$x\text{-intercept}$ is 5.

Step 4: Locate the vertical asymptotes. Determine the behaviour of the graph on either side of each vertical asymptote.

vertical asymptote none.

hole at $\left(-6,-11\right)$.

Step 5: Locate the horizontal or oblique asymptote. Determine points, if any, at which the graph of $R$ intersects this asymptote.

Here no horizontal or oblique asymptote.

Step 6: Graph $R$ using a graphing utility.

Step 7: