which of the following are vertical asymptotes of the function y = 3cot(2x) - 4? check all that apply. a. x…

Explanation:

Step1: Recall cotangent asymptote formula

The vertical - asymptotes of the cotangent function (y = A\cot(Bx - C)+D) occur at (Bx - C=k\pi), where (k\in\mathbb{Z}). For the function (y = 3\cot(2x)-4), we set (2x=k\pi), (k\in\mathbb{Z}). Then (x=\frac{k\pi}{2}), (k\in\mathbb{Z}).

Step2: Test each option

• Option A: When (x = \frac{\pi}{3}), (2x=\frac{2\pi}{3}\neq k\pi) for any integer (k).
• Option B: When (x=\pi), (2x = 2\pi=k\pi) where (k = 2).
• Option C: When (x = 2\pi), (2x=4\pi=k\pi) where (k = 4).
• Option D: When (x=\pm\frac{\pi}{2}), (2x=\pm\pi=k\pi) where (k=\pm1).

Answer:

B. (x = \pi), C. (x = 2\pi), D. (x=\pm\frac{\pi}{2})