Question

question
find the equation of all vertical asymptotes of the following function.
f(x)=\frac{2 x-6}{x^{2}-4 x-12}

Explanation:

Step1: Factor numerator and denominator

Numerator: \(2x - 6=2(x - 3)\)
Denominator: \(x^{2}-4x - 12=(x - 6)(x + 2)\)
So \(f(x)=\frac{2(x - 3)}{(x - 6)(x + 2)}\)

Step2: Find vertical asymptotes

Vertical asymptotes occur where the denominator is zero (and numerator is non - zero).
Set denominator \((x - 6)(x + 2)=0\), then \(x=6\) or \(x=-2\)
Check numerator at \(x = 6\): \(2(6 - 3)=6
eq0\)
Check numerator at \(x=-2\): \(2(-2 - 3)=-10
eq0\)

Answer:

Two Vertical Asymptotes (\(x = 6\) and \(x=-2\))