Question

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

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

Short answer

The graph for the function $R\left(x\right)=\frac{x}{\left(x-1\right)\left(x+2\right)}$is,

Step-by-step solution

Step 1 Given function is,

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

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

Also, $R$is in the lowest terms.

The $x$- intercept can be found by determining the real zeros of the numerator of $R$which is written in the lowest terms at $x=0$.

Here, the value of $R$is $0$when $x=0$. Hence, the$y$- intercept is,$0$.

Step 2 We can find the vertical asymptotes by finding the real zeros of the denominator of the function R.

The vertical asymptotes are,$x=\left\{1,-2\right\}$.

There is no horizontal asymptote of the graph. So, let us find the oblique asymptote.

This can be found by dividing the numerator by the denominator and the part except the remainder is the asymptote.

$y=\frac{x}{\left(x-1\right)\left(x+2\right)}$

Hence, the given polynomial has neither oblique nor horizontal asymptote.

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