Question

6.9.4 quiz: more than one vertical asymptote
question 6 of 10
how many vertical asymptotes does the graph of this function have?
$f(x)=\frac{5}{3x(x + 1)(x - 7)}$
a. 2
b. 3
c. 0
d. 1

Explanation:

Step1: Find the values that make the denominator zero

Set \(3x(x + 1)(x-7)=0\).
Using the zero - product property \(ab = 0\) implies \(a=0\) or \(b = 0\).
For \(3x(x + 1)(x - 7)=0\), we have:

• \(3x=0\Rightarrow x = 0\)
• \(x + 1=0\Rightarrow x=-1\)
• \(x-7=0\Rightarrow x = 7\)

Step2: Determine the number of vertical asymptotes

A vertical asymptote occurs at \(x=a\) if \(a\) makes the denominator of a rational function \(y=\frac{f(x)}{g(x)}\) (\(f(x)\) and \(g(x)\) are polynomials) zero and \(f(a)
eq0\).
Here, the numerator \(f(x)=5
eq0\) for \(x = 0\), \(x=-1\), and \(x = 7\).

Answer:

B. 3