To determine the domain of the function [tex]\( y = \cos^{-1} x \)[/tex], also known as the inverse cosine or arccosine function, we need to understand the nature of the cosine function and its inverse.
The inverse cosine function [tex]\( \cos^{-1} x \)[/tex] returns the angle whose cosine is [tex]\( x \)[/tex]. By definition, the cosine function, [tex]\( \cos(\theta) \)[/tex], maps any angle [tex]\( \theta \in [0, \pi] \)[/tex] to the range [tex]\([-1, 1]\)[/tex].
Therefore, for the inverse to exist, [tex]\( x \)[/tex] must lie within the range of the cosine function. Hence, the domain of [tex]\( \cos^{-1} x \)[/tex] must be all the [tex]\( x \)[/tex] values for which cosine can produce valid outputs. The valid values of [tex]\( x \)[/tex] are those where [tex]\(\cos(\theta) \)[/tex] produces results, which are [tex]\( [-1, 1] \)[/tex].
So, the domain of [tex]\( y = \cos^{-1} x \)[/tex] is [tex]\( [-1, 1] \)[/tex].
Therefore, the correct answer is:
[tex]\[ [-1, 1] \][/tex]
