find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{x^{2}+8 x-9}{2 x^{2}-12 x}$

Explanation:

Step1: Factor numerator and denominator

Numerator: (x^{2}+8x - 9=(x + 9)(x-1)) Denominator: (2x^{2}-12x=2x(x - 6)) So (f(x)=\frac{(x + 9)(x - 1)}{2x(x - 6)})

Step2: Find values that make denominator zero

Set (2x(x - 6)=0), we get (x = 0) or (x=6)

Step3: Check if these values make numerator zero

For (x = 0): ((0 + 9)(0 - 1)=-9\neq0) For (x = 6): ((6 + 9)(6 - 1)=75\neq0)

Answer:

The vertical asymptotes are (x = 0) and (x=6)