2 mark for review which of the following describes the graph of f(x) = cot x? a the graph has vertical…

2 mark for review which of the following describes the graph of f(x) = cot x? a the graph has vertical asymptotes at x = π/2 + πk, where k is any integer, and the function is increasing on all intervals in its domain. b the graph has vertical asymptotes at x = π/2 + πk, where k is any integer, and the function is decreasing on all intervals in its domain. c the graph has vertical asymptotes at x = π + πk, where k is any integer, and the function is increasing on all intervals in its domain. d the graph has vertical asymptotes at x = π + πk, where k is any integer, and the function is decreasing on all intervals in its domain.

2 mark for review which of the following describes the graph of f(x) = cot x? a the graph has vertical asymptotes at x = π/2 + πk, where k is any integer, and the function is increasing on all intervals in its domain. b the graph has vertical asymptotes at x = π/2 + πk, where k is any integer, and the function is decreasing on all intervals in its domain. c the graph has vertical asymptotes at x = π + πk, where k is any integer, and the function is increasing on all intervals in its domain. d the graph has vertical asymptotes at x = π + πk, where k is any integer, and the function is decreasing on all intervals in its domain.

Answer

Explanation:

Step1: Recall cotangent function properties

The cotangent function $y = \cot x=\frac{\cos x}{\sin x}$. Vertical - asymptotes occur where $\sin x = 0$.

Step2: Find vertical - asymptote values

We know that $\sin x=0$ when $x = k\pi$, where $k\in\mathbb{Z}$.

Step3: Determine the behavior of the cotangent function

The cotangent function $y = \cot x$ is decreasing on each interval of its domain $\left(k\pi,(k + 1)\pi\right)$ for $k\in\mathbb{Z}$.

Answer:

B. The graph has vertical asymptotes at $x = k\pi$, where $k$ is any integer, and the function is decreasing on all intervals in its domain.