Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x + 2}{x(x + 9)} ). (type integers or simplified fractions. use a comma to separate answers as needed type each answer only once.) c. the graph will cross the ( x )-axis at ( x=-2 ). (type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.) d. the function has no ( x )-intercept. determine the vertical asymptote(s). if any exist(s). select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. 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.) 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 one vertical asymptote, (type an equation. use integers or fractions for any numbers in the equation.) d. the function has no vertical asymptote.

Explanation:

Step1: Find the vertical asymptotes

For a rational function \(R(x)=\frac{f(x)}{g(x)}\), vertical asymptotes occur at the values of \(x\) that make \(g(x) = 0\).
Here, \(g(x)=x(x + 9)\). Set \(g(x)=0\), so \(x(x + 9)=0\).
Using the zero - product property \(ab = 0\) implies \(a = 0\) or \(b=0\).
If \(x=0\) or \(x+9 = 0\) (i.e., \(x=-9\)).

Answer:

B. The function has two vertical asymptotes. The leftmost asymptote is \(x=-9\), and the rightmost asymptote is \(x = 0\).