The domain of `f(x) = sqrt(cos^(-1)((1-|x|)/3))` is

Options

A. `[-4,0]`
B. `[0,2//sqrt(3)]`
C. `[-4,4]`
D. `[-3,3]`

Answer

To find the domain of the function \( f(x) = \sqrt{\cos^{-1}\left(\frac{1 - |x|}{3}\right)} \), we need to ensure that the expression inside the square root is non-negative and that the argument of the \(\cos^{-1}\) function lies within its defined range.

### Step 1: Determine the range of the \(\cos^{-1}\) function
The function \(\cos^{-1}(y)\) is defined for \(y\) in the interval \([-1, 1]\). Therefore, we need to ensure that:
\[ -1 \leq \frac{1 - |x|}{3} \leq 1 \]

### Step 2: Solve the inequalities
We will solve the inequalities step by step.

#### Inequality 1:
\[ \frac{1 - |x|}{3} \geq -1 \]
Multiplying both sides by 3:
\[ 1 - |x| \geq -3 \]
Rearranging gives:
\[ -|x| \geq -4 \implies |x| \leq 4 \]

#### Inequality 2:
\[ \frac{1 - |x|}{3} \leq 1 \]
Multiplying both sides by 3:
\[ 1 - |x| \leq 3 \]
Rearranging gives:
\[ -|x| \leq 2 \implies |x| \geq -2 \]
Since \(|x|\) is always non-negative, this inequality is always satisfied.

### Step 3: Combine the results
From the first inequality, we found that \(|x| \leq 4\). Thus, we can express this in terms of \(x\):
\[ -4 \leq x \leq 4 \]

### Step 4: Conclusion
The domain of the function \( f(x) \) is:
\[ \text{Domain} = [-4, 4] \]