If `[.]` denotes the greatest integer less than or equal to x and (.) denotes the least integer greater than or equal to x, then domain of the function `f(x)=sin^(-1){x+[x]+(x)}` is

Options

A. `x in {0}`
B. `x in [0,1]`
C. `x in (-1,0]`
D. `x in phi`

Explanation

To find the domain of the function \( f(x) = \sin^{-1}(x + [x] + (x)) \), where \([x]\) denotes the greatest integer less than or equal to \(x\) and \((x)\) denotes the least integer greater than or equal to \(x\), we need to ensure that the argument of the sine inverse function lies within its valid range, which is \([-1, 1]\).

### Step-by-step Solution:

1. **Understanding the Functions**:
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- The least integer function \((x)\) gives the smallest integer greater than or equal to \(x\).
- For any real number \(x\), we have \((x) = [x] + 1\) if \(x\) is not an integer, and \((x) = [x]\) if \(x\) is an integer.

2. **Expressing \(f(x)\)**:
- For integer \(x\):
\[ f(x) = \sin^{-1}(x + [x] + [x]) = \sin^{-1}(3x) \]
- For non-integer \(x\):
\[ f(x) = \sin^{-1}(x + [x] + ([x] + 1)) = \sin^{-1}(x + 2[x] + 1) \]

3. **Finding the Domain**:
- The domain of \(f(x)\) requires:
\[ -1 \leq x + [x] + (x) \leq 1 \]
- This can be simplified into two inequalities:
1. \(x + [x] + (x) \geq -1\)
2. \(x + [x] + (x) \leq 1\)

4. **Case 1: When \(x\) is an integer**:
- Let \(x = n\) (where \(n\) is an integer):
\[ f(n) = \sin^{-1}(3n) \]
- The condition becomes:
\[ -1 \leq 3n \leq 1 \]
- This implies:
\[ -\frac{1}{3} \leq n \leq \frac{1}{3} \]
- The only integer satisfying this is \(n = 0\).

5. **Case 2: When \(x\) is not an integer**:
- Let \(x = n + d\) where \(n\) is an integer and \(0 < d < 1\):
\[ f(x) = \sin^{-1}(n + d + 2n + 1) = \sin^{-1}(3n + d + 1) \]
- The condition becomes:
\[ -1 \leq 3n + d + 1 \leq 1 \]
- This simplifies to:
\[ -2 \leq 3n + d \leq 0 \]
- The first part gives:
\[ d \geq -2 - 3n \]
- The second part gives:
\[ d \leq -3n \]
- Since \(d\) is between \(0\) and \(1\), we analyze possible values of \(n\):
- For \(n = -1\): \(d\) must satisfy \(1 \leq d \leq 0\) (no solution).
- For \(n = 0\): \(0 \leq d \leq 0\) (only \(d = 0\) is valid).
- For \(n = 1\): \(d\) must satisfy \(0 \leq d \leq -3\) (no solution).

6. **Conclusion**:
- The only valid solution occurs when \(x = 0\).
- Therefore, the domain of the function \(f(x)\) is:
\[ \{0\} \]