Let's consider the function [tex]\( y = 3 \cot \left( \frac{1}{2} x \right) - 4 \)[/tex]. We need to determine where this function has vertical asymptotes. A vertical asymptote occurs where the argument of the cotangent function is undefined.
The cotangent function, [tex]\( \cot(\theta) \)[/tex], is undefined where [tex]\( \theta = n\pi \)[/tex], where [tex]\( n \)[/tex] is any integer.
We are dealing with [tex]\( \cot \left( \frac{1}{2} x \right) \)[/tex]. Therefore, we need to find [tex]\( x \)[/tex] such that:
[tex]\[ \frac{1}{2} x = n\pi \][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[ x = 2n\pi \][/tex]
This tells us that the function will have vertical asymptotes at [tex]\( x = 2n\pi \)[/tex] for [tex]\( n \)[/tex] being any integer.
Let's check each given option:
A. [tex]\( x = \pm 2\pi \)[/tex]:
- For [tex]\( n = 1 \)[/tex], [tex]\( x = 2(1)\pi = 2\pi \)[/tex]
- For [tex]\( n = -1 \)[/tex], [tex]\( x = 2(-1)\pi = -2\pi \)[/tex]
Thus, both [tex]\( x = 2\pi \)[/tex] and [tex]\( x = -2\pi \)[/tex] fit the form [tex]\( x = 2n\pi \)[/tex]. Hence, [tex]\( x = \pm 2\pi \)[/tex] are vertical asymptotes. This option is correct.
B. [tex]\( x = \frac{\pi}{2} \)[/tex]:
- If we set [tex]\( \frac{1}{2} x = \frac{\pi}{2} \)[/tex], solving for [tex]\( x \)[/tex] would give [tex]\( x = \pi \)[/tex], which does not fit the form [tex]\( 2n\pi \)[/tex].
Thus, this option is incorrect.
C. [tex]\( x = 3\pi \)[/tex]:
- If we set [tex]\( x = 3\pi \)[/tex], dividing both sides by [tex]\( 2 \)[/tex] gives [tex]\( \frac{1}{2} x = \frac{3\pi}{2} \)[/tex], which is not an integer multiple of [tex]\( \pi \)[/tex].
Thus, [tex]\( x = 3\pi \)[/tex] does not fit the form [tex]\( 2n\pi \)[/tex] and this option is incorrect.
D. [tex]\( x = 0 \)[/tex]:
- For [tex]\( n = 0 \)[/tex], [tex]\( x = 2(0)\pi = 0 \)[/tex].
Thus, [tex]\( x = 0 \)[/tex] fits the form [tex]\( 2n\pi \)[/tex]. Hence, this option is correct.
Combining the valid choices, the correct answers are:
A. [tex]\( x = \pm 2\pi \)[/tex]
D. [tex]\( x = 0 \)[/tex]
So, the vertical asymptotes of the function are given by the options A and D.
