Let `f:(2,4)->(1,3)` where `f(x) = x-[x/2]` (where [.] denotes the greatest integer function).Then `f^-1 (x)` is

Options

A. `2x`
B. `x+[x/2]`
C. `x+1`
D. `x-1`

Explanation

To find the inverse of the function \( f(x) = x - \left[\frac{x}{2}\right] \) where \( f: (2, 4) \to (1, 3) \), we will follow these steps:

### Step 1: Understand the Function
The function is defined as:
\[ f(x) = x - \left[\frac{x}{2}\right] \]
where \([\cdot]\) denotes the greatest integer function (floor function).

### Step 2: Determine the Range of \( f(x) \)
First, we need to evaluate \( \left[\frac{x}{2}\right] \) for \( x \) in the interval \( (2, 4) \):
- For \( x \in (2, 4) \):
- When \( 2 < x < 4 \), \( \frac{x}{2} \) will be in the interval \( (1, 2) \).
- Therefore, \( \left[\frac{x}{2}\right] = 1 \) because the greatest integer less than \( \frac{x}{2} \) in this range is 1.

### Step 3: Simplify the Function
Now substituting back into the function:
\[ f(x) = x - 1 \]
This means that for \( x \in (2, 4) \):
\[ f(x) = x - 1 \implies f: (2, 4) \to (1, 3) \]

### Step 4: Find the Inverse Function
To find the inverse function \( f^{-1}(x) \), we will set \( y = f(x) \):
\[ y = x - 1 \]
Now, solve for \( x \):
\[ x = y + 1 \]
Thus, we have:
\[ f^{-1}(x) = x + 1 \]

### Step 5: Determine the Domain of \( f^{-1}(x) \)
Since \( f(x) \) maps \( (2, 4) \) to \( (1, 3) \), the inverse function \( f^{-1}(x) \) will map \( (1, 3) \) back to \( (2, 4) \).

### Final Result
The inverse function is:
\[ f^{-1}(x) = x + 1 \quad \text{for } x \in (1, 3) \]