Question

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

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

Short answer

The graph is

Step-by-step solution

Given information

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

Calculate the domain

Setting denominator to zero, we get

$$\begin{gathered} x{(x-4)}^2=0 \\ x=0,4 \end{gathered}$$

The domain is all real numbers except$0,4$

Calculate the x intercept 

Setting numerator to zero, we get

$$\begin{gathered} (x-1)(x+2)(x-3)=0 \\ x=1,-2,3 \end{gathered}$$

There is no$y$intercept

Calculate the vertical asymptote

The zeroes of denominator is $x=0,4$so the vertical asymptote is$x=0,4$

Calculate the horizontal asymptote

The degree of numerator is same as denominator so the horizontal asymptote is

$$\begin{gathered} y=\frac{a}{b} \\ y=\frac{1}{1} \\ y=1 \end{gathered}$$

$$\begin{gathered} R\left(x\right)=\frac{x^3-2x^2-5x+16}{x^3-8x^2+16} \\ =1 \\ {x}^3-2x^2-5x+16=x^3-8x^2+16 \\ 6x^2-5x=0 \\ x=\frac{5}{6} \end{gathered}$$

It intersects at$(\frac{5}{6},1)$

Sketching the graph

The graph is