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 under the square root is defined and non-negative.

### Step 1: Determine the condition for the square root
The square root function is defined when its argument is non-negative. Therefore, we need:
\[ \cos^{-1}\left(\frac{1 - |x|}{2}\right) \geq 0 \]
Since the range of the inverse cosine function, \( \cos^{-1}(y) \), is \( [0, \pi] \), this condition is always satisfied for valid inputs of \( y \).

### Step 2: Determine the domain of \( \cos^{-1}(y) \)
The function \( \cos^{-1}(y) \) is defined for \( y \) in the interval \( [-1, 1] \). Thus, we require:
\[ -1 \leq \frac{1 - |x|}{2} \leq 1 \]

### Step 3: Solve the inequalities
We will solve the two inequalities separately.

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

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

### Step 4: Combine the results
From the first inequality, we have:
\[ |x| \leq 3 \]
This means:
\[ -3 \leq x \leq 3 \]

### Conclusion
Thus, the domain of the function \( f(x) \) is:
\[ [-3, 3] \]

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

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