Find the domain of `f(x)=1/(sqrt(x-[x]))` (b) `f(x)=1/(log[x])` `f(x)=log{x}`

Explanation

To find the domain of the given functions, we will analyze each function step by step.

### (a) \( f(x) = \frac{1}{\sqrt{x - [x]}} \)

1. **Understanding the function**: The function involves a square root in the denominator. For the function to be defined, the expression inside the square root must be positive:
\[ x - [x] > 0 \]
where \([x]\) is the greatest integer less than or equal to \(x\).

2. **Analyzing \(x - [x]\)**: The term \(x - [x]\) represents the fractional part of \(x\), denoted as \(\{x\}\). This fractional part is defined as:
\[ \{x\} = x - [x] \]
The fractional part \(\{x\}\) is always in the range \(0 \leq \{x\} < 1\).

3. **Setting the condition**: For the square root to be defined and positive:
\[ \{x\} > 0 \]
This means that \(x\) cannot be an integer, as for integer values, \(\{x\} = 0\).

4. **Conclusion for domain**: Therefore, the domain of \(f(x)\) is:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \mathbb{Z} \]
(all real numbers except integers).

### (b) \( f(x) = \frac{1}{\log[x]} \)

1. **Understanding the function**: The function involves a logarithm in the denominator. For the function to be defined, the logarithm must be positive:
\[ \log[x] > 0 \]

2. **Analyzing the logarithm**: The logarithm is positive when its argument is greater than 1:
\[ [x] > 1 \]
This means that \(x\) must be greater than 1 but less than 2, or greater than or equal to 2.

3. **Setting the conditions**: The integer part \([x]\) is equal to 1 when \(1 \leq x < 2\) and is equal to 2 when \(x \geq 2\). Thus, we need to exclude the interval where \([x] = 1\):
\[ x \notin [1, 2) \]

4. **Conclusion for domain**: Therefore, the domain of \(f(x)\) is:
\[ \text{Domain of } f(x) = (2, \infty) \]

### (c) \( f(x) = \log\{x\} \)

1. **Understanding the function**: The function involves the logarithm of the fractional part of \(x\). For this function to be defined, the fractional part must be positive:
\[ \{x\} > 0 \]

2. **Analyzing the fractional part**: As established earlier, the fractional part \(\{x\}\) is defined as:
\[ \{x\} = x - [x] \]
and is in the range \(0 < \{x\} < 1\).

3. **Setting the condition**: The condition \(\{x\} > 0\) means \(x\) cannot be an integer.

4. **Conclusion for domain**: Therefore, the domain of \(f(x)\) is:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \mathbb{Z} \]
(all real numbers except integers).

### Summary of Domains
- Domain of \( f(x) = \frac{1}{\sqrt{x - [x]}} \) is \( \mathbb{R} \setminus \mathbb{Z} \)
- Domain of \( f(x) = \frac{1}{\log[x]} \) is \( (2, \infty) \)
- Domain of \( f(x) = \log\{x\} \) is \( \mathbb{R} \setminus \mathbb{Z} \)