To determine which functions have asymptotes located at the values [tex]\( x = \pm n\pi \)[/tex], where [tex]\( n \)[/tex] is any integer, we need to analyze each function provided.
Option I: [tex]\( y = \csc x \)[/tex]
The cosecant function, [tex]\( y = \csc x \)[/tex], is defined as [tex]\( y = \frac{1}{\sin x} \)[/tex]. The sine function, [tex]\( \sin x \)[/tex], equals zero at [tex]\( x = n\pi \)[/tex] where [tex]\( n \)[/tex] is an integer. Therefore, [tex]\( \csc x \)[/tex] has vertical asymptotes at these points because dividing by zero is undefined. Hence, [tex]\( y = \csc x \)[/tex] has vertical asymptotes at [tex]\( x = n\pi \)[/tex].
Option II: [tex]\( y = \cos x \)[/tex]
The cosine function, [tex]\( y = \cos x \)[/tex], is a continuous function and does not have any vertical asymptotes. It only oscillates between -1 and 1 and never becomes undefined. Thus, [tex]\( y = \cos x \)[/tex] does not have vertical asymptotes at [tex]\( x = n\pi \)[/tex].
Option III: [tex]\( y = \tan x \)[/tex]
The tangent function, [tex]\( y = \tan x \)[/tex], is defined as [tex]\( y = \frac{\sin x}{\cos x} \)[/tex]. The function [tex]\( \cos x \)[/tex] equals zero at [tex]\( x = (2n + 1)\frac{\pi}{2} \)[/tex], where [tex]\( n \)[/tex] is an integer. Therefore, [tex]\( \tan x \)[/tex] has vertical asymptotes at these points, not at [tex]\( x = n\pi \)[/tex]. Hence, [tex]\( y = \tan x \)[/tex] does not have vertical asymptotes at [tex]\( x = \pm n\pi \)[/tex].
Option IV: [tex]\( y = \cot x \)[/tex]
The cotangent function, [tex]\( y = \cot x \)[/tex], is defined as [tex]\( y = \frac{\cos x}{\sin x} \)[/tex]. The sine function, [tex]\( \sin x \)[/tex], equals zero at [tex]\( x = n\pi \)[/tex] where [tex]\( n \)[/tex] is an integer, causing the cotangent function to be undefined at these points. Thus, [tex]\( y = \cot x \)[/tex] has vertical asymptotes at [tex]\( x = n\pi \)[/tex].
Based on this analysis:
- [tex]\( y = \csc x \)[/tex] (Option I) has vertical asymptotes at [tex]\( x = n\pi \)[/tex].
- [tex]\( y = \cot x \)[/tex] (Option IV) also has vertical asymptotes at [tex]\( x = n\pi \)[/tex].
Therefore, the correct answer is:
D. I and IV only
