which functions graph has asymptotes located ...
which functions graph has asymptotes located at the values x = ±nπ? i. y = csc x ii. y = cos x iii. y = tan x iv. y = cot x a. i and iii only b. ii only c. i and iv only d. i only
Answer
# Explanation:
## Step1: Recall csc x definition
$\csc x=\frac{1}{\sin x}$. Asymptotes occur where $\sin x = 0$. Since $\sin(n\pi)=0$ for all integers $n$, $y = \csc x$ has asymptotes at $x=\pm n\pi$.
## Step2: Analyze cos x
The function $y=\cos x$ has no vertical - asymptotes. Its range is $[- 1,1]$, and it is a continuous function.
## Step3: Analyze tan x
The function $y = \tan x=\frac{\sin x}{\cos x}$. Its asymptotes occur where $\cos x=0$, i.e., $x=\pm\frac{(2n + 1)\pi}{2},n\in\mathbb{Z}$, not at $x=\pm n\pi$.
## Step4: Analyze cot x
The function $y=\cot x=\frac{\cos x}{\sin x}$. Asymptotes occur where $\sin x = 0$, i.e., $x=\pm n\pi$.
# Answer:
C. I and IV only