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 the conditions that must be satisfied for the function to be defined.

### Step 1: Understand the components of the function
The function contains the floor function \( [x] \), which represents the greatest integer less than or equal to \( x \). The expression \( x - [x] \) gives us the fractional part of \( x \), denoted as \( \{x\} \).

### Step 2: Rewrite the function
We can rewrite the function as:
\[ f(x) = \frac{1}{\sqrt{\{x\}}} \]
where \( \{x\} = x - [x] \).

### Step 3: Determine the range of the fractional part
The fractional part \( \{x\} \) takes values in the interval \( [0, 1) \). Specifically:
- \( \{x\} = 0 \) when \( x \) is an integer.
- \( \{x\} \) is positive for non-integer values of \( x \).

### Step 4: Set conditions for the square root
For the function \( f(x) \) to be defined, the expression inside the square root must be positive:
\[ \{x\} > 0 \]
This means \( x - [x] > 0 \), which is true when \( x \) is not an integer.

### Step 5: Identify the domain
Since \( \{x\} \) is zero for all integer values of \( x \), the function \( f(x) \) will be undefined at those points. Therefore, the domain of \( f(x) \) consists of all real numbers except the integers.

Thus, the domain of the function \( f(x) \) can be expressed as:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \mathbb{Z} \]
where \( \mathbb{Z} \) represents the set of all integers.

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

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