Find the domain of definitions of the functions
(Read the symbols [*] and {*} as greatestintegers and fractional part functions respectively.)
`f(x) = sqrt(log _(x) (cos 2pi x))`

Answer

To find the domain of the function \( f(x) = \sqrt{\log_x(\cos(2\pi x))} \), we need to analyze the conditions under which the expression inside the square root is defined and non-negative.

### Step-by-Step Solution:

1. **Condition for Square Root**:
The expression inside the square root must be non-negative:
\[ \log_x(\cos(2\pi x)) \geq 0 \]
This implies that:
\[ \cos(2\pi x) > 0 \]

2. **Condition for Logarithm**:
The base of the logarithm \( x \) must satisfy the following conditions:
- \( x > 0 \)
- \( x \neq 1 \)

3. **Analyzing \( \cos(2\pi x) > 0 \)**:
The cosine function is positive in the intervals:
\[ 2\pi x \in (2k\pi, (2k+1)\pi) \quad \text{for } k \in \mathbb{Z} \]
This translates to:
\[ k < x < k + \frac{1}{2} \quad \text{for } k \in \mathbb{Z} \]

4. **Finding Specific Intervals**:
- For \( k = 0 \): \( 0 < x < \frac{1}{2} \)
- For \( k = 1 \): \( 1 < x < \frac{3}{2} \)
- For \( k = 2 \): \( 2 < x < \frac{5}{2} \)
- Continuing this pattern, we find intervals for \( k \geq 0 \).

5. **Combining Conditions**:
We must also ensure \( x \neq 1 \). Therefore, we exclude \( x = 1 \) from the interval \( (0, \frac{1}{2}) \) and \( (1, \frac{3}{2}) \).

6. **Final Domain**:
The domain of \( f(x) \) can be expressed as:
\[ (0, 1) \cup (1, \frac{3}{2}) \cup (2, \frac{5}{2}) \cup \ldots \]
In interval notation, this can be written as:
\[ (0, 1) \cup (1, \infty) \quad \text{excluding intervals where } \cos(2\pi x) \leq 0. \]

### Final Answer:
The domain of \( f(x) \) is:
\[ (0, 1) \cup (1, \infty) \]