Question

$y = \sqrt[3]{\tan x}$ $\qquad x$

Answer 1

The domain of the function is

$$\{x\in\mathbb{R} : x\neq 0 \text{ and } x\neq \frac{\pi}{2}+k\pi,\; k\in\mathbb{Z}\}.$$

Answer 2

Explanation:
For the function

$$y=\sqrt[3]{\frac{\tan x}{x}},$$

the cube root is defined for all real numbers. Thus, restrictions come from the expression inside the cube root:

1. The denominator $x$ must not be zero (i.e. $x \neq 0$).
2. $\tan x$ is undefined when $\cos x = 0$; that is, for $x=\frac{\pi}{2}+k\pi$ where $k\in\mathbb{Z}$.

Answer:
The domain of the function is

$$\{x\in\mathbb{R} : x\neq 0 \text{ and } x\neq \frac{\pi}{2}+k\pi,\; k\in\mathbb{Z}\}.$$