The domain of the function `f(x)=4sqrt(cos^(-1)((1-|x|)/(2)))` is

Options

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

Answer

To find the domain of the function \( f(x) = 4 \sqrt{\cos^{-1}\left(\frac{1 - |x|}{2}\right)} \), we need to ensure that the expression inside the square root is non-negative, as the square root function is defined for non-negative values.

### Step 1: Determine the condition for the inverse cosine function
The function \( \cos^{-1}(y) \) is defined for \( y \) in the range \([-1, 1]\). Therefore, we need to find the values of \( x \) for which:
\[ -1 \leq \frac{1 - |x|}{2} \leq 1 \]

### Step 2: Solve the inequalities
We will break this down into two inequalities.

#### Inequality 1:
\[ \frac{1 - |x|}{2} \geq -1 \]
Multiplying both sides by 2 gives:
\[ 1 - |x| \geq -2 \]
Rearranging this gives:
\[ |x| \leq 3 \]

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

### Step 3: Combine the results
From the first inequality, we have:
\[ |x| \leq 3 \]
This means:
\[ -3 \leq x \leq 3 \]

### Step 4: Determine the domain
The domain of the function \( f(x) \) is therefore:
\[ [-3, 3] \]

### Final Answer
Thus, the domain of the function \( f(x) = 4 \sqrt{\cos^{-1}\left(\frac{1 - |x|}{2}\right)} \) is:
\[ [-3, 3] \]