Find the domain of definitions of the functions
(Read the symbols [*] and {*} as greatestintegers and fractional part functions respectively.)
`f(x) = sqrt((5x - 6-x^(2)) [{ln {x}}]) + sqrt((7x -5-2x^(2))) + (ln ((7)/(2) - x))^(-1)`

Answer

To find the domain of the function

\[ f(x) = \sqrt{(5x - 6 - x^2) \lfloor \ln{x} \rfloor} + \sqrt{(7x - 5 - 2x^2)} + \left( \ln{\left(\frac{7}{2} - x\right)} \right)^{-1} \]

we need to analyze each component of the function to determine where it is defined.

### Step 1: Analyze the first term \(\sqrt{(5x - 6 - x^2) \lfloor \ln{x} \rfloor}\)

1. **Condition for the square root to be defined**:
\[ (5x - 6 - x^2) \lfloor \ln{x} \rfloor \geq 0 \]
This means both \(5x - 6 - x^2 \geq 0\) and \(\lfloor \ln{x} \rfloor \geq 0\) must hold true.

2. **Solve \(5x - 6 - x^2 \geq 0\)**:
Rearranging gives:
\[ -x^2 + 5x - 6 \geq 0 \implies x^2 - 5x + 6 \leq 0 \]
Factoring:
\[ (x - 2)(x - 3) \leq 0 \]
The solution to this inequality is:
\[ x \in [2, 3] \]

3. **Condition for \(\lfloor \ln{x} \rfloor \geq 0\)**:
This implies:
\[ \ln{x} \geq 0 \implies x \geq 1 \]

Combining these two conditions, we have:
\[ x \in [2, 3] \]

### Step 2: Analyze the second term \(\sqrt{(7x - 5 - 2x^2)}\)

1. **Condition for the square root to be defined**:
\[ 7x - 5 - 2x^2 \geq 0 \]
Rearranging gives:
\[ -2x^2 + 7x - 5 \geq 0 \implies 2x^2 - 7x + 5 \leq 0 \]
Factoring:
\[ (2x - 5)(x - 1) \leq 0 \]
The solution to this inequality is:
\[ x \in [1, \frac{5}{2}] \]

### Step 3: Analyze the third term \((\ln{\left(\frac{7}{2} - x\right)})^{-1}\)

1. **Condition for the logarithm to be defined**:
\[ \frac{7}{2} - x > 0 \implies x < \frac{7}{2} \]

### Step 4: Combine all conditions

Now we need to combine the intervals obtained from each part:

- From the first term, we have \(x \in [2, 3]\).
- From the second term, we have \(x \in [1, \frac{5}{2}]\).
- From the third term, we have \(x < \frac{7}{2}\).

The intersection of these intervals is:
\[ x \in [2, \frac{5}{2}] \]

### Final Domain

Thus, the final domain of the function \(f(x)\) is:
\[ \boxed{[2, \frac{5}{2}]} \]