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

Options

A. ` [0, 3]`
B. `[-6, 6]`
C. `[-1, 1]`
D. `[-3, 3]`

Explanation

To find the domain of the function

\[ f(x) = \sin^{-1} \left( \frac{2 - |x|}{4} \right) + \cos^{-1} \left( \frac{2 - |x|}{4} \right) + \tan^{-1} \left( \frac{2 - |x|}{4} \right), \]

we need to determine the constraints on \( x \) for each of the inverse functions involved.

### Step 1: Determine the domain for \( \sin^{-1} \left( \frac{2 - |x|}{4} \right) \)

The function \( \sin^{-1}(a) \) is defined for \( -1 \leq a \leq 1 \). Therefore, we need:

\[ -1 \leq \frac{2 - |x|}{4} \leq 1. \]

**Hint:** Start by solving the inequalities separately.

### Step 2: Solve the inequalities

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

2. **For the right inequality:**
\[ \frac{2 - |x|}{4} \leq 1 \]
Multiply both sides by 4:
\[ 2 - |x| \leq 4 \]
Rearranging gives:
\[ |x| \geq -2. \]

Since \( |x| \) is always non-negative, this condition is always satisfied.

**Hint:** Combine the results from both inequalities.

### Step 3: Combine results

From the inequalities, we have:
\[ |x| \leq 6. \]

This means:
\[ -6 \leq x \leq 6. \]

### Step 4: Determine the domain for \( \cos^{-1} \left( \frac{2 - |x|}{4} \right) \)

The domain for \( \cos^{-1}(a) \) is also \( -1 \leq a \leq 1 \). The analysis is identical to that of \( \sin^{-1} \), leading to the same condition:

\[ -6 \leq x \leq 6. \]

**Hint:** Notice that both inverse functions have the same domain constraints.

### Step 5: Determine the domain for \( \tan^{-1} \left( \frac{2 - |x|}{4} \right) \)

The function \( \tan^{-1}(a) \) is defined for all real numbers. Therefore, there are no additional constraints from this part of the function.

### Step 6: Final domain

The overall domain of \( f(x) \) is determined by the intersection of the domains of the three functions. Since both \( \sin^{-1} \) and \( \cos^{-1} \) give the domain \( -6 \leq x \leq 6 \) and \( \tan^{-1} \) is valid for all \( x \), the final domain of \( f(x) \) is:

\[ \text{Domain of } f(x) = [-6, 6]. \]

### Conclusion

Thus, the domain of the function \( f(x) \) is:

\[ \boxed{[-6, 6]}. \]