find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{3 x^{2}-3 x}{2 x-2}$
find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{3 x^{2}-3 x}{2 x-2}$
Answer
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\neq1)
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})