Question

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

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

Short answer

The graph is

Step-by-step solution

Given information

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

Factoring the numerator and denominator

Factoring, we get

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

Calculate the domain

Setting denominator to zero, we get

$$\begin{gathered} x-4=0 \\ x=4 \end{gathered}$$

The domain is all real numbers except$4$

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$R\left(0\right)=3$i.e.$(0,3)$

Calculate the vertical asymptote

Setting denominator to zero, we get

$$\begin{gathered} x-4=0 \\ x=4 \end{gathered}$$

Calculating the horizontal asymptote

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

The oblique asymptote is$y=x+5$

Sketching the graph

The graph is