Question

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

$F\left(x\right)=\frac{x^2-3x-4}{x+2}$

Short answer

The graph is

Step-by-step solution

Given information

Given function$F\left(x\right)=\frac{x^2-3x-4}{x+2}$

Factoring the numerator and denominator

Factoring, we get

$$\begin{gathered} F\left(x\right)=\frac{x^2-3x-4}{x+2} \\ F\left(x\right)=\frac{x^2-4x+x-4}{x+2} \\ F\left(x\right)=\frac{x(x-4)+1(x-4)}{x+2} \\ F\left(x\right)=\frac{(x+1)(x-4)}{(x+2)} \end{gathered}$$

Calculating the domain

Setting denominator to zero, we get

$$\begin{gathered} x+2=0 \\ x=-2 \end{gathered}$$

The domain is all real numbers except$-2$

Calculating the x intercept 

Setting numerator to zero, we get

$$\begin{gathered} (x+1)(x-4)=0 \\ x=-1,4 \end{gathered}$$

The$y$intercept is$F\left(0\right)=-2$i.e.$(0,-2)$

Calculating the vertical asymptote

Setting denominator to zero, we get

$x=-2$

Calculate the horizontal asymptote

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

The oblique asymptote is$y=x-5$

Sketching the graph

The graph is