Question

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

1. Find the domain of $R$.
2. Find the$x$-intercepts of$R$.

Short answer

1. The domain of $R$is,$\left\{x|x\ne 2\right\}$.
2. The$x$-intercepts of$R$is$x=0$.

Step-by-step solution

Part a Step 1 Given equation is,

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

First find the zeros of the denominator.

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

The domain of $R$is the set of all the real numbers except when$x=2$.

Part b Step 2 Substitute Rx=0 in the given equation.

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

As$2$is not in the domain, the value of$x$-intercept is$0$.