Question

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

$R\left(x\right)=\frac{3}{(x-1)\left(x^2-4\right)}$

Short answer

The graph is

Step-by-step solution

Given information

Given function$R\left(x\right)=\frac{3}{(x-1)\left(x^2-4\right)}$

Factor the numerator and denominator

Factoring, we get

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

Calculating the domain

Setting the denominator to zero, we get

$$\begin{gathered} (x-1)(x-2)(x+2)=0 \\ x=1,2,-2 \end{gathered}$$

The domain is all real numbers except$1,2,-2$

Calculate the x intercept

Calculating, we get

$3$not equal to zero

So no $x$intercept and $y$intercept is

$R\left(0\right)=\frac{3}{4}$

Calculating the vertical asymptote

Setting denominator to zero, we get

$$\begin{gathered} (x-1)(x+2)(x+2)=0 \\ x=1,-2,2 \end{gathered}$$is the vertical asymptote

Calculating the horizontal asymptote

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

The intersection point is $$\begin{gathered} R\left(x\right)=\frac{3}{x^3-x^2-4x+4} \\ =0 \end{gathered}$$

So it doesn't intersects$R$

Sketching the graph

The graph is