Question 4 of 25 Find part of the domain of the tangent function \( f(x)=\tan x \) that lies in the set \( [0,2 \pi) \). (Use symbolic notation and fractions where needed. Give your answer for domain as intervals in the form ( \( * \), \( * \) ). Use the symbol © for infinity, \( U \) for combining intervals, and an appropriate type of parenthesis " \( ( \) " " ")", "[" or "]" depending on whether the interval is open or closed.) domain of \( f \) : \( \square \)

The tangent function \( f(x) = \tan x \) is undefined where the cosine of \( x \) is zero, which occurs at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) within the interval \( [0, 2\pi) \). Therefore, these points must be excluded from the domain. The domain of \( f(x) = \tan x \) within \( [0, 2\pi) \) is all real numbers in this interval except \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). This can be expressed as the union of three intervals: \[ \text{domain of } f : [0, \tfrac{\pi}{2}) \, U \, (\tfrac{\pi}{2}, \tfrac{3\pi}{2}) \, U \, (\tfrac{3\pi}{2}, 2\pi) \] **Answer:** domain of \( f \) : \([0, \tfrac{\pi}{2}) \, U \, (\tfrac{\pi}{2}, \tfrac{3\pi}{2}) \, U \, (\tfrac{3\pi}{2}, 2\pi)\)