which functions graph has asymptotes located at the values x = ±nπ? i. y = csc x ii. y = cos x iii. y = tan…

Answer:

A. I and IV only

Explanation:

Step1: Recall the definition of cosecant function

$y = \csc x=\frac{1}{\sin x}$. Asymptotes occur when $\sin x = 0$. Since $\sin x=0$ when $x = n\pi$, $n\in\mathbb{Z}$, $y = \csc x$ has asymptotes at $x = n\pi$.

Step2: Recall the properties of cosine function

$y=\cos x$ has no vertical - asymptotes. Its range is $[- 1,1]$, and it is a continuous function for all real $x$.

Step3: Recall the properties of tangent function

$y = \tan x=\frac{\sin x}{\cos x}$. Asymptotes occur when $\cos x = 0$, i.e., $x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}$, not at $x = n\pi$.

Step4: Recall the definition of cotangent function

$y=\cot x=\frac{\cos x}{\sin x}$. Asymptotes occur when $\sin x = 0$, i.e., $x = n\pi$, $n\in\mathbb{Z}$. So $y=\cot x$ has asymptotes at $x = n\pi$.