Question

find all vertical asymptotes of the following function.
$f(x) = \frac{3x - 21}{4x^2 - 9}$
answer attempt 2 out of 4
no vertical asymptotes
no vertical asymptotes
one vertical asymptote
two vertical asymptotes

Explanation:

Step1: Factor denominator

The denominator is \(4x^2 - 9\), which is a difference of squares. So, \(4x^2 - 9=(2x - 3)(2x + 3)\).

Step2: Find undefined points

A function is undefined where the denominator is zero. Set \((2x - 3)(2x + 3)=0\).
Solving \(2x - 3 = 0\) gives \(x=\frac{3}{2}\), and solving \(2x + 3 = 0\) gives \(x=-\frac{3}{2}\).

Step3: Check numerator at these points

The numerator is \(3x - 21\). At \(x=\frac{3}{2}\), \(3(\frac{3}{2})-21=\frac{9}{2}-21=\frac{9 - 42}{2}=-\frac{33}{2}
eq0\). At \(x = -\frac{3}{2}\), \(3(-\frac{3}{2})-21=-\frac{9}{2}-21=\frac{-9 - 42}{2}=-\frac{51}{2}
eq0\). So, these are vertical asymptotes.

Answer:

Two Vertical Asymptotes