To determine which of the given points is an asymptote of the function [tex]\( y = \sec(x) \)[/tex], we need to understand where [tex]\(\sec(x)\)[/tex], which is defined as [tex]\( \sec(x) = \frac{1}{\cos(x)} \)[/tex], becomes undefined. This occurs when the denominator, [tex]\(\cos(x)\)[/tex], is zero because division by zero is undefined.
The cosine function [tex]\(\cos(x)\)[/tex] is zero at the following values:
[tex]\[ x = \frac{\pi}{2} + k\pi \][/tex]
where [tex]\( k \)[/tex] is any integer. We will check each provided option to see if it satisfies this condition.
1. Option [tex]\( x = -2\pi \)[/tex]:
[tex]\[ \cos(-2\pi) = \cos(2\pi) = 1 \][/tex]
Since [tex]\(\cos(-2\pi) \neq 0\)[/tex], there is no asymptote at [tex]\( x = -2\pi \)[/tex].
2. Option [tex]\( x = -\frac{\pi}{6} \)[/tex]:
[tex]\[ \cos\left(-\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \][/tex]
Since [tex]\(\cos\left(-\frac{\pi}{6}\right) \neq 0\)[/tex], there is no asymptote at [tex]\( x = -\frac{\pi}{6} \)[/tex].
3. Option [tex]\( x = \pi \)[/tex]:
[tex]\[ \cos(\pi) = -1 \][/tex]
Since [tex]\(\cos(\pi) \neq 0\)[/tex], there is no asymptote at [tex]\( x = \pi \)[/tex].
4. Option [tex]\( x = \frac{3\pi}{2} \)[/tex]:
[tex]\[ \cos\left(\frac{3\pi}{2}\right) = 0 \][/tex]
Since [tex]\(\cos\left(\frac{3\pi}{2}\right) = 0\)[/tex], there is an asymptote at [tex]\( x = \frac{3\pi}{2} \)[/tex].
Therefore, [tex]\( x = \frac{3\pi}{2} \)[/tex] is an asymptote of [tex]\( y = \sec(x) \)[/tex].
The correct answer is:
[tex]\( x = \frac{3\pi}{2} \)[/tex]
