which of the following are vertical asymptote...
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. \(x=\pm\frac{\pi}{2}\)