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

which of the following are vertical asymptotes of the function y = 3cot(2x) - 4? check all that apply. a. x = π b. x = π/3 c. x = 2π d. x = ±π/2

which of the following are vertical asymptotes of the function y = 3cot(2x) - 4? check all that apply. a. x = π b. x = π/3 c. x = 2π d. x = ±π/2

Answer

Explanation:

Step1: Recall cotangent asymptote formula

The vertical - asymptotes of the cotangent function (y = \cot(u)) occur when (u = n\pi), where (n\in\mathbb{Z}) (the set of all integers). For the function (y = 3\cot(2x)-4), we set (2x=n\pi).

Step2: Solve for (x)

Solving (2x = n\pi) for (x) gives (x=\frac{n\pi}{2}), (n\in\mathbb{Z}).

Step3: Check each option

  • Option A: When (n = 2), (x=\frac{2\pi}{2}=\pi).
  • Option B: If (x=\frac{\pi}{3}), then (\frac{n\pi}{2}=\frac{\pi}{3}), and (n=\frac{2}{3}\notin\mathbb{Z}).
  • Option C: When (n = 4), (x=\frac{4\pi}{2}=2\pi).
  • Option D: When (n=\pm1), (x = \pm\frac{\pi}{2}).

Answer:

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