Question

6.9.4 quiz: more than one vertical asymptote
at which values of x does the function f(x) have a vertical asymptote? check all that apply.
f(x)=\frac{3}{(x + 2)(x - 7)}
a. 2
b. 7
c. -2
d. -7
e. -14
f. 0

Explanation:

Step1: Recall the definition of vertical asymptote

A vertical asymptote occurs at \(x = a\) if the denominator of a rational function is zero at \(x = a\) and the numerator is non - zero at \(x = a\).

Step2: Set the denominator equal to zero

Set \((x + 2)(x - 7)=0\).
Using the zero - product property \(ab = 0\) implies \(a=0\) or \(b = 0\).
If \(x+2=0\), then \(x=-2\).
If \(x - 7=0\), then \(x = 7\).

Step3: Check the numerator

The numerator of \(F(x)=\frac{3}{(x + 2)(x - 7)}\) is \(3\). Since \(3
eq0\) for all \(x\), the function has vertical asymptotes at \(x=-2\) and \(x = 7\).

Answer:

B. \(7\), C. \(-2\)