Question

question
find all vertical asymptotes of the following function
f(x)=\frac{3 x^{2}-108}{x-9}
answer attempt 1 out of 2
no vertical asymptotes

Explanation:

Step1: Simplify the function

First, factor the numerator \(3x^{2}-108 = 3(x^{2}-36)=3(x + 6)(x - 6)\).
The function \(f(x)=\frac{3x^{2}-108}{x - 9}=\frac{3(x + 6)(x - 6)}{x - 9}\).

Step2: Find the vertical asymptote

Vertical asymptotes occur where the denominator is zero (and the numerator is non - zero at that point).
Set the denominator \(x-9=0\), then \(x = 9\).
When \(x = 9\), the numerator \(3(9 + 6)(9 - 6)=3\times15\times3=135
eq0\).

Answer:

The vertical asymptote is \(x = 9\).