Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x^{2}+5 x - 24}{x^{2}-x - 6} ).
determine the vertical asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has one vertical asymptote, ( x=-2 ).
(type an equation. use integers or fractions for any numbers in the equation.)
b. the function has two vertical asymptotes. the leftmost asymptote is , and the rightmost asymptote is .
(type equations. use integers or fractions for any numbers in the equations.)
c. the function has three vertical asymptotes. the leftmost asymptote is , the middle asymptote is , and the rightmost asymptote is .
(type equations. use integers or fractions for any numbers in the equations.)
d. the function has no vertical asymptote.
determine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. there is a hole in the graph at the point .
(type an ordered pair using integers or fractions.)
b. there are no holes in the graph.

Explanation:

Step1: Factor numerator and denominator

$$\begin{align*} x^{2}+5x - 24&=(x + 8)(x-3)\\ x^{2}-x - 6&=(x - 3)(x + 2) \end{align*}$$

Step2: Simplify the rational function

$$R(x)=\frac{(x + 8)(x - 3)}{(x - 3)(x + 2)}=\frac{x + 8}{x + 2},x eq3$$

Since \(x = 3\) makes both numerator and denominator zero (a removable discontinuity - a hole), and \(x=-2\) makes the denominator zero (vertical asymptote)

Answer:

A. The function has one vertical asymptote, \(x=-2\)
A. There is a hole in the graph at the point \((3,\frac{11}{5})\)