Question

find all vertical asymptotes of the following function.

$f(x)=\frac{3 x^{2}-3 x}{2 x-2}$

Explanation:

Step1: Simplify the function

Factor the numerator and denominator.
Numerator: \(3x^{2}-3x = 3x(x - 1)\)
Denominator: \(2x-2=2(x - 1)\)
So \(f(x)=\frac{3x(x - 1)}{2(x - 1)}=\frac{3x}{2},x
eq1\)

Step2: Analyze the vertical asymptote

A vertical asymptote occurs where the function is undefined (after simplification, when the original denominator is zero and the numerator is non - zero at that point).
For the original function \(y = \frac{3x^{2}-3x}{2x - 2}\), the denominator \(2x-2 = 0\) gives \(x = 1\). But after simplification \(f(x)=\frac{3x}{2}\) (a linear function), and when \(x = 1\), the original function has a removable discontinuity (a hole) rather than a vertical asymptote.

Answer:

There are no vertical asymptotes for the function \(f(x)=\frac{3x^{2}-3x}{2x - 2}\)