Chapter 4.3, Problem 18AYU

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

$R\left(x\right)=\frac{x^2+x-12}{x^2-4}$

To determine

To find: Ananlyze the given function.

Answer to Problem 18AYU

Explanation of Solution

Given:

$R\left(x\right) = \frac{x^2 + x - 12}{x^2 - 4}$

Step 1: Factor $\text{R}$ to get

$R\left(x\right) = \frac{x^2 + x - 12}{x^2 - 4}$

That is $x = -2, x = 2$.

The domain $\left\{x\text{|}x \ne -2, x \ne 2\right\}$

Step 2: $\text{R}$ is in lowest terms.

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

$y\text{-intercept}$ 3.

$x\text{-intercept}$ is $-4$ and 3.

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

$x = -2, x = 2$

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

Here horizontal asymptote $y = 1$

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

Step 7: