If `f(x)=(x^(2)-[x^(2)])/(x^(2)-[x^(2)-2])` (where, `[.]` represents the greatest integer part of x), then the range of f(x) is

Options

A. `[0, 1)`
B. `(-1, 1)`
C. `(0, oo)`
D. `[0, (1)/(3))`

Answer

To find the range of the function \( f(x) = \frac{x^2 - [x^2]}{x^2 - [x^2 - 2]} \), where \([.]\) denotes the greatest integer function (or floor function), we will analyze the behavior of the function step by step.

### Step 1: Understand the components of the function
The function \( f(x) \) consists of two parts: the numerator \( x^2 - [x^2] \) and the denominator \( x^2 - [x^2 - 2] \).

- The term \( [x^2] \) represents the greatest integer less than or equal to \( x^2 \).
- The term \( [x^2 - 2] \) represents the greatest integer less than or equal to \( x^2 - 2 \).

### Step 2: Analyze the numerator
The numerator \( x^2 - [x^2] \) gives the fractional part of \( x^2 \), denoted as \( \{x^2\} \). This value will always be in the range \( [0, 1) \).

### Step 3: Analyze the denominator
For the denominator \( x^2 - [x^2 - 2] \):
- If \( n = [x^2] \), then \( n \leq x^2 < n + 1 \).
- Therefore, \( [x^2 - 2] = n - 2 \) if \( n \geq 2 \) and \( [x^2 - 2] = -1 \) if \( n = 1 \) or \( n = 0 \).

Thus, we can express the denominator as:
- If \( n \geq 2 \): \( x^2 - (n - 2) = x^2 - n + 2 \)
- If \( n = 1 \): \( x^2 - (-1) = x^2 + 1 \)
- If \( n = 0 \): \( x^2 - (-1) = x^2 + 1 \)

### Step 4: Determine the range of \( f(x) \)
Now we will analyze \( f(x) \) in different intervals based on the integer values of \( n \):

1. **For \( n = 0 \) (i.e., \( 0 \leq x^2 < 1 \)):**
- \( f(x) = \frac{x^2}{x^2 + 1} \)
- As \( x^2 \) approaches \( 1 \), \( f(x) \) approaches \( \frac{1}{2} \).

2. **For \( n = 1 \) (i.e., \( 1 \leq x^2 < 2 \)):**
- \( f(x) = \frac{x^2 - 1}{x^2 + 1} \)
- As \( x^2 \) approaches \( 2 \), \( f(x) \) approaches \( \frac{1}{3} \).

3. **For \( n = 2 \) (i.e., \( 2 \leq x^2 < 3 \)):**
- \( f(x) = \frac{x^2 - 2}{x^2 - 1} \)
- As \( x^2 \) approaches \( 3 \), \( f(x) \) approaches \( \frac{1}{2} \).

4. **For \( n \geq 3 \):**
- The function will continue to behave similarly, but we need to check the limits as \( x^2 \) increases.

### Step 5: Conclusion
From the analysis, we can see that the function \( f(x) \) will vary between \( 0 \) and \( 1 \) depending on the intervals defined by the integer parts of \( x^2 \).

Thus, the range of \( f(x) \) is \( [0, 1) \).