Question

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

$R\left(x\right)=\frac{6}{x^2-x-6}$

Short answer

The graph for the function $R\left(x\right)=\frac{6}{x^2-x-6}$is,

Step-by-step solution

Step 1 Given function is,

$R\left(x\right)=\frac{6}{x^2-x-6}$

which may also written as,

$R\left(x\right)=\frac{6}{\left(x+2\right)\left(x-3\right)}$

The domain of$R$is,$\left\{x|x\ne -2,3\right\}$

Step 2 R is in lowest terms.

The $x$- intercept can be determined by finding the real zeros of the numerator of $R$which is in the lowest terms.

Therefore, by solving we have no real zeros.

Hence, there is no $x$- intercept. The $y$- intercept can be found by determining the value of $R$when $x=0$and the value for $R$is $-1$when $x=0$. Hence, the $y$- intercept is,$-1$.

Step 3 we can find the vertical asymptotes by finding the zeroes of the denomination of the rational function in lowest terms.

So, the vertical asymptotes are, $x=\left\{-2,3\right\}$.

There is no horizontal asymptote of the graph. So, let us calculate the oblique asymptotes. This can be found by dividing the numerator by denominator and the part except the remainder is the asymptote.

In $R$when we do this procedure, we can find,

$y=\frac{2x+4}{x-1}$

Therefore, the polynomial has neither horizontal nor oblique asymptote.

Step 4 Now plot a graph by using the known values.