Question

In Problems $7-44$, follow Steps 1 through 7 on page 228 to analyze the graph of each function.

$R\left(x\right)=\frac{x^2-x-12}{x+5}$

Short answer

The graph is

Step-by-step solution

Given information

Given function$R\left(x\right)=\frac{x^2-x-12}{x+5}$

Factoring the numerator and denominator

Factoring, we get

$$\begin{gathered} R\left(x\right)=\frac{x^2-x-12}{x+5} \\ R\left(x\right)=\frac{x^2-4x+3x-12}{x+5} \\ R\left(x\right)=\frac{x(x-4)+3(x-4)}{x+5} \\ R\left(x\right)=\frac{(x+3)(x-4)}{(x+5)} \end{gathered}$$

Calculating the domain

On setting the denominator to zero,

$x=-5$

The domain is all real numbers except$x=-5$

Calculate the x intercept

Setting numerator to zero, we get

$$\begin{gathered} (x+3)(x-4)=0 \\ x=-3,4 \end{gathered}$$

The$y$intercept is$R\left(0\right)=-\frac{12}{5}$i.e.$(0,-\frac{12}{5})$

Calculating the vertical asymptote

The vertical asymptote is zeroes of denominator i.e.$x=-5$

Calculating the horizontal asymptote

The degree of numerator is greater than denominator so there is an oblique asymptote

$R\left(x\right)=x-6+\frac{18}{x+5}$

The oblique asymptote is$y=x-6$

Sketching the graph

The graph is