Question

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

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

Short answer

The graph for the function $R\left(x\right)=\frac{3}{x^2-4}$is,

Step-by-step solution

Step 1 Given function is,

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

which may also written as,

$R\left(x\right)=\frac{3}{\left(x+2\right)\left(x-2\right)}$

The domain of $R$is, $\left\{x|x\ne \pm 2\right\}$

As $0$is in the domain, substitute $x=0$in the given function.

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

So, the obtained point is,$\left(0,\frac{-3}{4}\right)$.

Step 2 Ris in the lowest terms.

The real zeros of the denominator are the real solutions of the equation.

$$\begin{gathered} \left(x+2\right)\left(x-2\right)=0 \\ x=\pm 2 \end{gathered}$$

The two lines $x=2,x=-2$are the two vertical asymptotes of the graph.

In the given function, degree of the numerator is less than the degree of the denominator. So, $R$is proper.

So, the line $y=0$is the horizontal asymptote of the graph.

Substitute $R\left(x\right)=0$in the given function.

$$\begin{gathered} 0=\frac{3}{x^2-4} \\ 0\ne 3 \end{gathered}$$

There exists no solution. So, the graph of$R$does not intersect the horizontal asymptote at$y=0$.

Step 3 Now plot a graph by using the known values.