question\nfind the equation of all vertical asymptotes of the following function.\n f(x)=\frac{2…

Explanation:

Step1: Find the domain of the function

For the function (f(x)=\frac{2x - 2}{\sqrt{-x^{2}+2x + 15}}), the denominator (\sqrt{-x^{2}+2x + 15}) must satisfy (-x^{2}+2x + 15>0). Solve the quadratic inequality (-x^{2}+2x + 15>0). First, solve the equation (-x^{2}+2x + 15 = 0). Using the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) for (ax^{2}+bx + c = 0) (here (a=-1), (b = 2), (c = 15)), we have (x=\frac{-2\pm\sqrt{4+60}}{-2}=\frac{-2\pm\sqrt{64}}{-2}=\frac{-2\pm8}{-2}). The roots are (x=-3) and (x = 5). The quadratic function (y=-x^{2}+2x + 15) is a parabola opening downwards ((a=-1<0)), so the solution of (-x^{2}+2x + 15>0) is (-3<x<5).

Step2: Analyze the vertical asymptotes

Vertical asymptotes occur where the function is undefined and the one - sided limits are infinite. Since the function (f(x)) is only defined for (-3<x<5) (the domain of (f(x)) is an open interval ((-3,5))), and the function is continuous on its domain (because the numerator (2x - 2) is a polynomial and the denominator (\sqrt{-x^{2}+2x + 15}) is non - zero and continuous on ((-3,5))).

Answer:

No Vertical Asymptotes