The domain of the function `f(x)=sqrt(cos^(- 1)((1-|x|)/2))` is

Options

A. `[-3, 3]`
B. `(-oo, -3) uu( 3, oo)`
C. `(-oo, -3] uu [3, oo)`
D. `1`

Explanation

To find the domain of the function \( f(x) = \sqrt{\cos^{-1}\left(\frac{1 - |x|}{2}\right)} \), we need to ensure that the expression inside the square root is defined and non-negative.

### Step-by-Step Solution:

1. **Identify the conditions for the square root:**
The expression inside the square root, \( \cos^{-1}\left(\frac{1 - |x|}{2}\right) \), must be non-negative. Since the range of \( \cos^{-1}(y) \) is from \( 0 \) to \( \pi \) for \( y \) in the interval \([-1, 1]\), we can conclude that \( \cos^{-1}(y) \) is always non-negative.

2. **Determine the conditions for \( \cos^{-1} \):**
For \( \cos^{-1}\left(\frac{1 - |x|}{2}\right) \) to be defined, the argument \( \frac{1 - |x|}{2} \) must lie within the range \([-1, 1]\):
\[ -1 \leq \frac{1 - |x|}{2} \leq 1 \]

3. **Solve the inequalities:**
- **Left inequality:**
\[ -1 \leq \frac{1 - |x|}{2} \]
Multiplying both sides by 2:
\[ -2 \leq 1 - |x| \]
Rearranging gives:
\[ |x| \leq 3 \]

- **Right inequality:**
\[ \frac{1 - |x|}{2} \leq 1 \]
Multiplying both sides by 2:
\[ 1 - |x| \leq 2 \]
Rearranging gives:
\[ |x| \geq -1 \]
Since the absolute value is always non-negative, this condition is always satisfied.

4. **Combine the results:**
The only relevant condition we have is \( |x| \leq 3 \). This translates to:
\[ -3 \leq x \leq 3 \]

5. **Conclusion:**
Therefore, the domain of the function \( f(x) \) is:
\[ \boxed{[-3, 3]} \]