The function [tex]\( y = \tan(x) \)[/tex] is defined for all real numbers except at points where the tangent function is undefined. The tangent function [tex]\( \tan(x) \)[/tex] is undefined at values where [tex]\( x = \frac{\pi}{2} + n\pi \)[/tex] for any integer [tex]\( n \)[/tex]. At these points, the cosine function in the denominator of [tex]\( \tan(x) = \frac{\sin(x)}{\cos(x)} \)[/tex] is equal to zero, causing the tangent function to be undefined.
Given this information, the domain of [tex]\( y = \tan(x) \)[/tex] is all real numbers where [tex]\( x \neq \frac{\pi}{2} + n\pi \)[/tex] for any integer [tex]\( n \)[/tex].
Therefore, the correct answer is:
C. all real numbers where [tex]\( x \neq \frac{\pi}{2} + \pi n \)[/tex]
