If `f(x) = log_([x-1])(|x|)/(x)`,where [.] denotes the greatest integer function,then

Options

A. `D(f) = [3, oo), R(f) = {0, 1} `
B. `D(f) = [ 3, oo), R(f) = [3, oo), R(f)= {0}`
C. `D(f) = (2, oo), R(f) = {0, 1}`
D. `D(f) = (3, oo), R(f) = {0}`

Explanation

To solve the problem, we need to analyze the function \( f(x) = \log_{[\text{greatest integer}(x) - 1]} \left( \frac{|x|}{x} \right) \), where \([\cdot]\) denotes the greatest integer function. We will determine the domain and range of this function step by step.

### Step 1: Determine the conditions for the logarithm
The logarithm \( \log_a(b) \) is defined under the following conditions:
1. The base \( a \) must be greater than 0 and not equal to 1.
2. The argument \( b \) must be greater than 0.

### Step 2: Analyze the base of the logarithm
The base of our logarithm is \( [x] - 1 \).
- For \( [x] - 1 > 0 \): This implies \( [x] > 1 \) or \( x \geq 2 \).
- For \( [x] - 1 \neq 1 \): This implies \( [x] \neq 2 \) or \( x \notin [2, 3) \).

Combining these two conditions, we find:
- \( x \) must be in the intervals \( [3, \infty) \).

### Step 3: Analyze the argument of the logarithm
The argument of the logarithm is \( \frac{|x|}{x} \).
- This expression is equal to 1 for \( x > 0 \) and -1 for \( x < 0 \).
- Since we are considering \( x \geq 3 \), \( \frac{|x|}{x} = 1 \) which is always greater than 0.

### Step 4: Combine the conditions for the domain
From the analysis above, the domain of \( f(x) \) is:
- \( x \in [3, \infty) \).

### Step 5: Determine the range of the function
Now we need to find the range of \( f(x) \):
- Since \( \frac{|x|}{x} = 1 \) for \( x \geq 3 \), we have:
\[ f(x) = \log_{[\text{greatest integer}(x) - 1]}(1) \]
- The logarithm of 1 in any base (as long as the base is valid) is 0:
\[ f(x) = 0 \quad \text{for } x \in [3, \infty) \]

### Conclusion
The function \( f(x) \) is constant and equal to 0 for all \( x \) in the domain. Therefore, the domain and range of the function are:
- **Domain**: \( [3, \infty) \)
- **Range**: \( \{0\} \) (a singleton set)

### Final Answer
- Domain: \( [3, \infty) \)
- Range: \( \{0\} \)