Question

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

$G\left(x\right)=\frac{3x}{x^2-1}$

Short answer

The graph is

Step-by-step solution

Given information

Given function$G\left(x\right)=\frac{3x}{x^2-1}$

Calculating the domain

Setting the denominator to zero, we get

$$\begin{gathered} x^2-1=0 \\ x=\pm 1 \end{gathered}$$

The domain is all real numbers except$1,-1$

Factoring the numerator and denominator

Factoring, we get

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

Calculate the x intercept

Setting denominator to zero, we get

$$\begin{gathered} 3x=0 \\ x=0 \end{gathered}$$

$G\left(0\right)=0$

Calculate the vertical asymptote

Setting to zero, we get

$$\begin{gathered} x^2-1=0 \\ {x}^2=1 \\ x=\pm 1 \end{gathered}$$is the vertical asymptote

Calculating the horizontal asymptote

Here degree of numerator is less than denominator so the asymptote is $y=0$

$$\begin{gathered} G\left(x\right)=\frac{3x}{x^2-1} \\ =0 \\ x=0 \end{gathered}$$

So the intersection point is$(0,0)$

Sketching the graph

The graph is