Domain of `f(x)=sin^(-1)(([x])/({x}))`, where `[*]` and `{*}` denote greatest integer and fractional parts.

Answer

To find the domain of the function \( f(x) = \sin^{-1}\left(\frac{[x]}{\{x\}}\right) \), where \([x]\) denotes the greatest integer function and \(\{x\}\) denotes the fractional part function, we need to ensure that the argument of the \(\sin^{-1}\) function lies within the interval \([-1, 1]\).

### Step-by-Step Solution:

1. **Understanding the Functions**:
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- The fractional part function \(\{x\} = x - [x]\) gives the non-integer part of \(x\), which lies in the interval \([0, 1)\).

2. **Setting Up the Inequalities**:
- For \(f(x)\) to be defined, we need:
\[ -1 \leq \frac{[x]}{\{x\}} \leq 1 \]
- This means we need to analyze the cases when \(\{x\} \neq 0\) since division by zero is undefined.

3. **Identifying the Cases**:
- The fractional part \(\{x\} = 0\) when \(x\) is an integer. Therefore, \(x\) cannot be an integer.
- Thus, we consider \(x\) in the intervals where it is not an integer.

4. **Case 1: \(x > 0\)**:
- If \(x\) is a positive non-integer, then \([x] \geq 0\) and \(0 < \{x\} < 1\).
- Thus, \(\frac{[x]}{\{x\}} \geq 0\).
- Since \(\{x\}\) is positive and less than 1, \(\frac{[x]}{\{x\}} > 1\) if \([x] \geq 1\).
- Therefore, for \(x \geq 1\), \(\frac{[x]}{\{x\}} > 1\), which is not valid.
- The only valid case is when \(0 < x < 1\), where \([x] = 0\) and \(\frac{[x]}{\{x\}} = 0\), which is valid since \(0 \in [-1, 1]\).

5. **Case 2: \(x < 0\)**:
- If \(x\) is negative and not an integer, then \([x] < -1\) (since the greatest integer less than a negative non-integer is the next lower integer).
- The fractional part \(\{x\}\) is still in the interval \((0, 1)\).
- Thus, \(\frac{[x]}{\{x\}} < -1\), which is not valid since it does not lie in the interval \([-1, 1]\).

6. **Conclusion**:
- The only valid interval for \(x\) is when \(0 < x < 1\).
- Therefore, the domain of the function \(f(x)\) is:
\[ \text{Domain of } f(x) = (0, 1) \]

### Summary of the Domain:
The domain of the function \( f(x) = \sin^{-1}\left(\frac{[x]}{\{x\}}\right) \) is \( (0, 1) \).