Question

determine the vertical asymptote(s) of the function. if none exist, state that fact.
f(x)=\frac{3 x}{x^{2}-9}
select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice
a. the function has one vertical asymptote. (type an equation)
b. the function has three vertical asymptotes. the leftmost asymptote is , the middle asymptote is , and the rightmost asymptote is
(type equations)
c. the function has two vertical asymptotes. the leftmost asymptote is and the rightmost asymptote is
(type equations)
d. the function has no vertical asymptotes

Explanation:

Step1: Find the denominator's zeros

Set the denominator \(x^{2}-9 = 0\). Using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), where \(a=x\) and \(b = 3\), we have \((x + 3)(x-3)=0\). Solving \(x+3 = 0\) gives \(x=-3\), and solving \(x - 3=0\) gives \(x = 3\).

Step2: Check for common factors

The numerator is \(3x\) and the denominator is \((x + 3)(x - 3)\). There are no common factors between the numerator and the denominator.

Answer:

C. The function has two vertical asymptotes. The leftmost asymptote is \(x=-3\) and the rightmost asymptote is \(x = 3\)