The domain of the real valued function `log_([x+1//2])|x^2 -x-2|` is

Options

A. `[3/2,2) U (2,oo)`
B. `[3/2 ,oo)-{2}`
C. `[1/2 ,oo)-{2}`
D. None

Answer

To find the domain of the function \( f(x) = \log_{\left[\frac{x+1}{2}\right]} |x^2 - x - 2| \), we need to consider the conditions for the logarithmic function and the absolute value function.

### Step 1: Identify the conditions for the logarithmic function

The logarithmic function \( \log_b(a) \) has the following conditions:
1. The base \( b \) must be greater than 0 and not equal to 1.
2. The argument \( a \) must be greater than 0.

### Step 2: Analyze the argument \( |x^2 - x - 2| \)

We need to find when \( |x^2 - x - 2| > 0 \):
- First, solve \( x^2 - x - 2 = 0 \) to find the points where the expression equals zero.

Factoring the quadratic:
\[ x^2 - x - 2 = (x - 2)(x + 1) = 0 \]
This gives us the roots:
\[ x = 2 \quad \text{and} \quad x = -1 \]

The expression \( x^2 - x - 2 \) will be zero at \( x = 2 \) and \( x = -1 \). Therefore, \( |x^2 - x - 2| > 0 \) for all \( x \) except \( x = 2 \) and \( x = -1 \).

### Step 3: Analyze the base \( \left[\frac{x+1}{2}\right] \)

Next, we need to ensure that the base \( \left[\frac{x+1}{2}\right] \) satisfies:
1. \( \left[\frac{x+1}{2}\right] > 0 \)
2. \( \left[\frac{x+1}{2}\right] \neq 1 \)

#### Condition 1: \( \left[\frac{x+1}{2}\right] > 0 \)

The greatest integer function \( \left[\frac{x+1}{2}\right] \) is greater than 0 when:
\[ \frac{x+1}{2} \geq 1 \implies x + 1 \geq 2 \implies x \geq 1 \]

#### Condition 2: \( \left[\frac{x+1}{2}\right] \neq 1 \)

To ensure that \( \left[\frac{x+1}{2}\right] \neq 1 \):
\[ \frac{x+1}{2} < 2 \implies x + 1 < 4 \implies x < 3 \]

### Step 4: Combine the conditions

From the analysis, we have:
1. \( x \geq 1 \)
2. \( x < 3 \)
3. Exclude \( x = 2 \) and \( x = -1 \)

Thus, the valid range for \( x \) is:
\[ 1 \leq x < 3 \quad \text{excluding} \quad x = 2 \]

### Step 5: Express the domain in interval notation

The domain can be expressed as:
\[ [1, 2) \cup (2, 3) \]

### Final Answer

The domain of the function \( f(x) = \log_{\left[\frac{x+1}{2}\right]} |x^2 - x - 2| \) is:
\[ [1, 2) \cup (2, 3) \]