The domain of the function `f(x)=(1)/(sqrt((x)-[x]))` where [*] denotes the greatest integer function is

Options

A. R
B. `R^(+)`
C. `R^(-)`
D. `R-Z`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{x - [x]}} \), where \([x]\) denotes the greatest integer function, we need to analyze the expression inside the square root and ensure that the function is defined.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). Therefore, we can express \(x\) as:
\[ x = [x] + \{x\} \]
where \(\{x\} = x - [x]\) is the fractional part of \(x\).

2. **Rewriting the Function**:
We can rewrite the function \(f(x)\) as:
\[ f(x) = \frac{1}{\sqrt{\{x\}}} \]
This means we need to analyze the fractional part \(\{x\}\).

3. **Conditions for the Function to be Defined**:
For the function \(f(x)\) to be defined:
- The denominator cannot be zero: \(\sqrt{\{x\}} \neq 0\).
- The expression inside the square root must be positive: \(\{x\} > 0\).

4. **Analyzing the Fractional Part**:
The fractional part \(\{x\}\) is defined as:
\[ \{x\} = x - [x] \]
The range of \(\{x\}\) is \(0 \leq \{x\} < 1\). Thus:
- \(\{x\} = 0\) when \(x\) is an integer.
- \(\{x\} > 0\) when \(x\) is not an integer.

5. **Conclusion about the Domain**:
Therefore, for \(f(x)\) to be defined, \(x\) must not be an integer. Thus, the domain of \(f(x)\) is:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \mathbb{Z} \]
This means the function is defined for all real numbers except integers.

### Final Answer:
The domain of the function \( f(x) = \frac{1}{\sqrt{x - [x]}} \) is all real numbers except integers, or in set notation:
\[ \text{Domain} = \{ x \in \mathbb{R} : x \notin \mathbb{Z} \} \]