Chapter 4.5, Problem 14AYU

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

$R\left(x\right)=\frac{x}{\left(x-1\right)\left(x+2\right)}$

To determine

To find: Analyze the graph of each function.

$R\left(x\right) = \frac{x}{\left(x - 1\right)\left(x + 2\right)}$

Answer to Problem 14AYU

Explanation of Solution

Given:

$R\left(x\right) = \frac{x}{\left(x - 1\right)\left(x + 2\right)}$

Calculation:

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

$R\left(x\right) = \frac{x}{\left(x - 1\right)\left(x + 2\right)}$

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

Step 2: $R$ is in lowest terms

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

$y\text{-intercepts}$ is 0

$x\text{-intercept}$ is 0

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

$x = -2, x = 1$.

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

Horizontal asymptote: $x = 0$, intersected at $\left(0, 0\right)$.

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

Step 7: