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

Options

A. [-6,6]
B. `(-oo,2) uu (2,3)`
C. `(2,3)`
D. `[-6,2) uu(2,3)`

Answer

To find the domain of the function \( f(x) = \cos^{-1}\left(\frac{2 - |x|}{4}\right) \), we need to ensure that the argument of the inverse cosine function lies within the interval \([-1, 1]\).

### Step-by-Step Solution:

1. **Set up the inequality**:
We need to ensure that:
\[ -1 \leq \frac{2 - |x|}{4} \leq 1 \]

2. **Solve the left side of the inequality**:
Start with the left side:
\[ -1 \leq \frac{2 - |x|}{4} \]
Multiply both sides by 4:
\[ -4 \leq 2 - |x| \]
Rearranging gives:
\[ |x| \leq 6 \]

3. **Solve the right side of the inequality**:
Now consider the right side:
\[ \frac{2 - |x|}{4} \leq 1 \]
Multiply both sides by 4:
\[ 2 - |x| \leq 4 \]
Rearranging gives:
\[ |x| \geq -2 \]
Since the absolute value is always non-negative, this condition is always satisfied.

4. **Combine the results**:
From the left side, we found \( |x| \leq 6 \). The condition \( |x| \geq -2 \) does not impose any additional restrictions. Therefore, we have:
\[ |x| \leq 6 \]

5. **Express the domain in terms of \( x \)**:
The inequality \( |x| \leq 6 \) translates to:
\[ -6 \leq x \leq 6 \]

### Conclusion:
The domain of the function \( f(x) = \cos^{-1}\left(\frac{2 - |x|}{4}\right) \) is:
\[ [-6, 6] \]