If `f(x) =1- x, x in [-3,3]`, then the domain of `fof (x)` is

Options

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

Explanation

To find the domain of the function \( f(f(x)) \) where \( f(x) = 1 - x \) and \( x \) is in the interval \([-3, 3]\), we can follow these steps:

### Step 1: Identify the function and its domain
The function given is:
\[ f(x) = 1 - x \]
The domain of \( f(x) \) is:
\[ x \in [-3, 3] \]

### Step 2: Determine the range of \( f(x) \)
To find the range of \( f(x) \) when \( x \) is in \([-3, 3]\), we can evaluate the function at the endpoints of the interval:
- When \( x = -3 \):
\[ f(-3) = 1 - (-3) = 1 + 3 = 4 \]
- When \( x = 3 \):
\[ f(3) = 1 - 3 = -2 \]

Thus, the range of \( f(x) \) is:
\[ f(x) \in [-2, 4] \]

### Step 3: Determine the domain of \( f(f(x)) \)
Next, we need to find \( f(f(x)) \):
\[ f(f(x)) = f(1 - x) = 1 - (1 - x) = x \]
This means \( f(f(x)) = x \) is valid as long as \( x \) is in the range of \( f(x) \).

### Step 4: Find the intersection of the range of \( f(x) \) and the domain of \( f(x) \)
The domain of \( f(f(x)) \) must be within the range of \( f(x) \) and also within the original domain of \( f(x) \):
- The range of \( f(x) \) is \([-2, 4]\).
- The original domain of \( f(x) \) is \([-3, 3]\).

To find the domain of \( f(f(x)) \), we take the intersection of these two intervals:
\[ [-2, 4] \cap [-3, 3] = [-2, 3] \]

### Conclusion
Thus, the domain of \( f(f(x)) \) is:
\[ [-2, 3] \]