Let's find the range of [tex]\( y = \sec^{-1}(x) \)[/tex].
First, recall that the secant function is defined as the reciprocal of the cosine function, i.e., [tex]\(\sec(y) = \frac{1}{\cos(y)}\)[/tex]. The domain of the secant function, [tex]\(\sec(y)\)[/tex], is where the cosine function is not zero. Therefore, [tex]\(\sec(y)\)[/tex] is defined for all [tex]\( y \)[/tex] where [tex]\( \cos(y) \neq 0 \)[/tex].
However, since we are interested in the inverse secant function, [tex]\( y = \sec^{-1}(x) \)[/tex], we need to find the range of this function given the domain of [tex]\( \sec(y) \)[/tex]. The function [tex]\( y = \sec^{-1}(x) \)[/tex] is the inverse of [tex]\( x = \sec(y) \)[/tex], so we need to determine where [tex]\( \sec(y) \)[/tex] can take on all possible inputs [tex]\( x \)[/tex].
The function [tex]\( \sec(y) \)[/tex] has a specific range. The function [tex]\( \sec(y) \)[/tex], being the reciprocal of [tex]\( \cos(y) \)[/tex], will be either [tex]\( x \geq 1 \)[/tex] or [tex]\( x \leq -1 \)[/tex] because [tex]\(\cos(y)\)[/tex] ranges between -1 and 1, except at zero where [tex]\(\sec(y)\)[/tex] is undefined. Specifically, this means:
[tex]\[ \sec(y) \geq 1 \quad \text{or} \quad \sec(y) \leq -1 \][/tex]
To make the function [tex]\( \sec^{-1}(x) \)[/tex] a proper inverse function, we restrict the range of [tex]\( y \)[/tex] to principal values for which the secant function is one-to-one. The principal values for the secant function are generally taken between:
[tex]\[ y \in [0, \frac{\pi}{2}) \cup [\frac{\pi}{2}, \pi] \][/tex]
Thus, the range of [tex]\( y = \sec^{-1}(x) \)[/tex] is:
[tex]\[ \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right] \][/tex]
This matches the last option:
[tex]\[ \left[0, \frac{\pi}{2}\right) \text{ and } \left(\frac{\pi}{2}, \pi\right] \][/tex]
So, the correct answer is:
[tex]\[
\left[0, \frac{\pi}{2}\right) \text{ and } \left(\frac{\pi}{2}, \pi\right]
\][/tex]
