Find the Range of function
`f(x) = (x- [x])/(1 - [x] + x) , [.]` = G.I.F.) :

Options

A. `[0,1]`
B. `[0, 1//2]`
C. `[1//2 , 1]`
D. `[0,1//2]`

Answer

To find the range of the function \( f(x) = \frac{x - [x]}{1 - [x] + x} \), where \([x]\) denotes the greatest integer function (G.I.F.), we can follow these steps:

### Step 1: Understand the components of the function
The function \( f(x) \) can be rewritten using the fractional part of \( x \). The fractional part of \( x \) is defined as:
\[ \{x\} = x - [x] \]
Thus, we can express \( f(x) \) as:
\[ f(x) = \frac{\{x\}}{1 - [x] + x} \]

### Step 2: Rewrite the denominator
We can express \( x \) in terms of its integer and fractional parts:
\[ x = [x] + \{x\} \]
Substituting this into the denominator gives:
\[ 1 - [x] + x = 1 - [x] + ([x] + \{x\}) = 1 + \{x\} \]
So, we can rewrite \( f(x) \) as:
\[ f(x) = \frac{\{x\}}{1 + \{x\}} \]

### Step 3: Analyze the range of the fractional part
The fractional part \( \{x\} \) varies between \( 0 \) and \( 1 \) (i.e., \( 0 \leq \{x\} < 1 \)).

### Step 4: Determine the behavior of \( f(x) \)
Now, we need to analyze the function \( f(x) = \frac{\{x\}}{1 + \{x\}} \) as \( \{x\} \) varies from \( 0 \) to \( 1 \).

- When \( \{x\} = 0 \):
\[ f(x) = \frac{0}{1 + 0} = 0 \]

- When \( \{x\} \) approaches \( 1 \):
\[ f(x) = \frac{1}{1 + 1} = \frac{1}{2} \]

### Step 5: Find the range
Since \( f(x) \) is continuous and strictly increasing in the interval \( [0, 1) \), the range of \( f(x) \) is:
\[ [0, \frac{1}{2}) \]

### Conclusion
Thus, the range of the function \( f(x) \) is:
\[ \text{Range of } f(x) = [0, \frac{1}{2}) \]