The function `f(x)=sqrt(|tan x|+tan x)/(sqrt3x)` is defined for:

Options

A. R
B. `R-{1/3}`
C. `R^(+)-{nx +pi/2|n in 1^(+)}`
D. None of these

Answer

To determine the domain of the function \( f(x) = \frac{\sqrt{|\tan x| + \tan x}}{\sqrt{3}x} \), we need to analyze the conditions under which this function is defined.

### Step 1: Analyze the square root condition
The expression inside the square root, \( |\tan x| + \tan x \), must be non-negative for the square root to be defined.

1. **Case 1**: When \( \tan x \geq 0 \) (i.e., \( \tan x \) is positive or zero):
- Here, \( |\tan x| = \tan x \).
- Thus, \( |\tan x| + \tan x = \tan x + \tan x = 2\tan x \geq 0 \).
- This condition is satisfied for \( \tan x \geq 0 \).

2. **Case 2**: When \( \tan x < 0 \) (i.e., \( \tan x \) is negative):
- Here, \( |\tan x| = -\tan x \).
- Thus, \( |\tan x| + \tan x = -\tan x + \tan x = 0 \).
- This condition is satisfied as well since the square root of zero is defined.

From both cases, we conclude that \( |\tan x| + \tan x \) is always non-negative.

### Step 2: Analyze the denominator condition
The denominator \( \sqrt{3}x \) must not be equal to zero:
- This means \( x \neq 0 \).

### Step 3: Analyze the tangent function
The tangent function \( \tan x \) is undefined at certain points:
- Specifically, \( \tan x \) is undefined when \( \cos x = 0 \), which occurs at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \).

### Step 4: Combine the conditions
From the analysis, we have:
- \( x \neq 0 \) (to avoid division by zero).
- \( x \neq \frac{\pi}{2} + n\pi \) (to avoid points where \( \tan x \) is undefined).

### Final Domain
Thus, the function \( f(x) \) is defined for:
\[ x \in \mathbb{R} \setminus \left\{ 0, \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} \]