Question

f(x)=\frac{2}{x^{2}-2x - 15}

choose the equation(s) below that are vertical asymptotes for the graph of the given function. select all that apply or indicate that there are no vertical asymptotes.

□ a. x = 15
□ b. x = 5
□ c. x = - 3
□ d. x = - 5
□ e. y = 1
□ f. x = 0
□ g. y = 0
□ h. there are no vertical asymptotes.

Explanation:

Step1: Find the denominator's roots

For the function \(f(x)=\frac{2}{x^{2}-2x - 15}\), factor the denominator \(x^{2}-2x - 15=(x - 5)(x+3)\).
Set the denominator equal to zero: \((x - 5)(x + 3)=0\).
Using the zero - product property \(ab = 0\) implies \(a=0\) or \(b = 0\), so \(x-5=0\) gives \(x = 5\) and \(x + 3=0\) gives \(x=-3\).

Step2: Determine vertical asymptotes

The vertical asymptotes of a rational function \(y=\frac{N(x)}{D(x)}\) (where \(N(x)\) is the numerator and \(D(x)\) is the denominator) occur at the values of \(x\) that make \(D(x)=0\) (provided \(N(x)
eq0\) at those values). Here, \(N(x) = 2
eq0\) for \(x = 5\) and \(x=-3\).

Answer:

B. \(x = 5\), C. \(x=-3\)