If [x] denote the greater integer less than or equal to x, then the domain of definition of the real valued funciton `f(c) =log_([x + 1//2])|x^(2) -x-2|`, is

Options

A. `[3//2, oo)`
B. `[3//2, 2)uu (2, oo)`
C. `(1//2, 2) uu (2, oo)`
D. none of these

Explanation

To find the domain of the function \( f(x) = \log_{[\frac{x + 1}{2}]} |x^2 - x - 2| \), we need to ensure that the conditions for the logarithm are satisfied. The conditions for a logarithmic function \( \log_b(y) \) are:

1. The base \( b > 0 \) and \( b \neq 1 \).
2. The argument \( y > 0 \).

Let's analyze the function step by step:

### Step 1: Analyze the base
The base of the logarithm is \( [\frac{x + 1}{2}] \).

- **Condition 1**: \( [\frac{x + 1}{2}] > 0 \)

This implies:
\[ \frac{x + 1}{2} \geq 1 \implies x + 1 \geq 2 \implies x \geq 1 \]

- **Condition 2**: \( [\frac{x + 1}{2}] \neq 1 \)

This occurs when:
\[ 1 \leq \frac{x + 1}{2} < 2 \]
Solving this gives:
\[ 1 \leq x + 1 < 4 \implies 0 \leq x < 3 \]
Thus, we have:
\[ x \in [0, 3) \]

### Step 2: Combine conditions for the base
From the conditions on the base, we have:
1. \( x \geq 1 \)
2. \( x < 3 \)

So, the combined condition for the base is:
\[ x \in [1, 3) \]

### Step 3: Analyze the argument
The argument of the logarithm is \( |x^2 - x - 2| \).

- We need \( |x^2 - x - 2| > 0 \).
- First, we find when \( x^2 - x - 2 = 0 \):
\[ x^2 - x - 2 = (x - 2)(x + 1) = 0 \]
This gives the roots \( x = 2 \) and \( x = -1 \).

The expression \( x^2 - x - 2 \) changes sign at these points. We analyze the intervals:
1. For \( x < -1 \): \( x^2 - x - 2 > 0 \)
2. For \( -1 < x < 2 \): \( x^2 - x - 2 < 0 \)
3. For \( x > 2 \): \( x^2 - x - 2 > 0 \)

Thus, \( |x^2 - x - 2| > 0 \) when:
- \( x < -1 \)
- \( x > 2 \)

### Step 4: Combine conditions for the argument
Since we are interested in the domain where both conditions for the base and argument hold, we need to find the intersection of:
1. \( x \in [1, 3) \) (from the base)
2. \( x \in (-\infty, -1) \cup (2, \infty) \) (from the argument)

The only overlapping interval is:
\[ x \in (2, 3) \]

### Final Domain
Thus, the domain of the function \( f(x) \) is:
\[ \boxed{(2, 3)} \]