Question

find all vertical asymptotes of the following function.
$f(x) = \frac{3x^2 - 27}{x^2 - 12x + 27}$
answer attempt 1 out of 2
no vertical asymptotes
no vertical asymptotes
one vertical asymptote
two vertical asymptotes

Explanation:

Step1: Factor numerator and denominator

Factor \(3x^2 - 27\): \(3(x^2 - 9)=3(x - 3)(x + 3)\)
Factor \(x^2 - 12x + 27\): \((x - 3)(x - 9)\)

Step2: Simplify the function

\(f(x)=\frac{3(x - 3)(x + 3)}{(x - 3)(x - 9)}\), cancel \((x - 3)\) (for \(x
eq3\)): \(f(x)=\frac{3(x + 3)}{x - 9}\)

Step3: Find vertical asymptotes

Vertical asymptotes occur where denominator is zero (and numerator non - zero) after simplification.
Set \(x - 9 = 0\), so \(x = 9\).
Check \(x = 3\): after canceling, the function is undefined at \(x = 3\) but it's a hole (removable discontinuity), not an asymptote. So there is one vertical asymptote.

Answer:

One Vertical Asymptote