Find the domain of following functions:
`f(x)=1/(sqrt(x-[x]))`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{x - [x]}} \), we need to analyze the expression inside the square root and ensure that it meets the necessary conditions for the function to be defined.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The term \([x]\) represents the greatest integer less than or equal to \(x\). The expression \(x - [x]\) gives us the fractional part of \(x\), denoted as \(\{x\}\). Thus, we can rewrite the function as:
\[ f(x) = \frac{1}{\sqrt{\{x\}}} \]

2. **Finding the Range of the Fractional Part**:
The fractional part \(\{x\}\) is defined as:
\[ \{x\} = x - [x] \]
The range of \(\{x\}\) is \(0 \leq \{x\} < 1\).

3. **Conditions for the Function to be Defined**:
For \(f(x)\) to be defined, the expression inside the square root must be positive:
\[ \{x\} > 0 \]
This means:
\[ x - [x] > 0 \]
This inequality holds true when \(x\) is not an integer. If \(x\) is an integer, then \(\{x\} = 0\) and the function becomes undefined since we cannot take the square root of zero in the denominator.

4. **Conclusion on the Domain**:
Therefore, the domain of the function \(f(x)\) is all real numbers except for integers. In interval notation, this can be expressed as:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \mathbb{Z} \]
or in words, the set of all real numbers excluding integers.

### Final Answer:
The domain of the function \( f(x) = \frac{1}{\sqrt{x - [x]}} \) is:
\[ \text{Domain} = \{ x \in \mathbb{R} \mid x \text{ is not an integer} \} \]