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What is the domain of [tex]y = \tan(x)[/tex]?

A. all real numbers where [tex]x \neq n\pi[/tex]
B. [tex](-\infty, \infty)[/tex]
C. all real numbers where [tex]x \neq \frac{\pi}{2} + \pi n[/tex]
D. [tex](0, \pi)[/tex]

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Sagot :

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]