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

Options

A. `[-6,3)`
B. `[-6,2)`
C. `[-6,3]`
D. none of these

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 range of \([-1, 1]\).

### Step-by-step Solution:

1. **Set up the inequality**:
The argument of the cosine inverse function must satisfy:
\[ -1 \leq \frac{2-x}{4} \leq 1 \]

2. **Solve the left side of the inequality**:
Start with the left part of the inequality:
\[ -1 \leq \frac{2-x}{4} \]
Multiply both sides by 4 (note that multiplying by a positive number does not change the inequality):
\[ -4 \leq 2 - x \]
Rearranging gives:
\[ -4 - 2 \leq -x \quad \Rightarrow \quad -6 \leq -x \]
Multiplying by -1 (which reverses the inequality):
\[ 6 \geq x \quad \Rightarrow \quad x \leq 6 \]

3. **Solve the right side of the inequality**:
Now consider the right part of the inequality:
\[ \frac{2-x}{4} \leq 1 \]
Again, multiply both sides by 4:
\[ 2 - x \leq 4 \]
Rearranging gives:
\[ -x \leq 4 - 2 \quad \Rightarrow \quad -x \leq 2 \]
Multiplying by -1 (which reverses the inequality):
\[ x \geq -2 \]

4. **Combine the results**:
From the two parts, we have:
\[ -2 \leq x \leq 6 \]
Thus, the domain of the function \( f(x) \) is:
\[ x \in [-2, 6] \]

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