Question

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

$G\left(x\right)=\frac{x}{x^2-4}$

Short answer

The graph is

Step-by-step solution

Given information

Given the function$G\left(x\right)=\frac{x}{x^2-4}$

Factoring the numerator and denominator

Factoring, we get

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

Calculate the domain

On setting the denominator to zero, we get

$x=2,-2$

So the domain is all real numbers except$2,-2$

Substitute numerator equal to zero to get the xintercept 

Substituting, we get

$x=0$

So the $x$intercept is zero so the$y$intercept is$0$

Calculate the vertical asymptote

Setting denominator to zero, we get

$$\begin{gathered} x^2-4=0 \\ x=2,-2 \end{gathered}$$is the vertical asymptote

Calculating the horizontal asymptote

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

The intersection point is $$\begin{gathered} G\left(x\right)=\frac{x}{x^2-4} \\ =0 \end{gathered}$$

So the point is$(0,0)$

Sketching the graph

The graph is