Let `f:(6, 8)rarr (9, 11)` be a function defined as `f(x)=x+[(x)/(2)]` (where `[.]` denotes the greatest integer function), then `f^(-1)(x)` is equal to

Options

A. `x-[(x)/(2)]`
B. `-x-3`
C. `x-3`
D. `(1)/(x+[(x)/(2)])`

Answer

To find the inverse of the function \( f(x) = x + \left[ \frac{x}{2} \right] \), where \([.]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Understand the function
The function \( f(x) = x + \left[ \frac{x}{2} \right] \) combines the value of \( x \) with the greatest integer less than or equal to \( \frac{x}{2} \).

### Step 2: Determine the range of \( f(x) \)
Given the domain \( (6, 8) \):
- For \( x = 6 \):
\[ f(6) = 6 + \left[ \frac{6}{2} \right] = 6 + 3 = 9 \]
- For \( x = 8 \):
\[ f(8) = 8 + \left[ \frac{8}{2} \right] = 8 + 4 = 12 \]
Thus, the range of \( f(x) \) for \( x \in (6, 8) \) is \( (9, 12) \).

### Step 3: Identify the intervals for \( f(x) \)
To find \( f^{-1}(x) \), we need to analyze the function in intervals:
1. For \( 6 < x < 8 \):
- \( \left[ \frac{x}{2} \right] = 3 \) when \( 6 < x < 8 \).
- Therefore, \( f(x) = x + 3 \).

### Step 4: Solve for \( x \) in terms of \( y \)
Let \( y = f(x) \), then:
\[ y = x + 3 \]
Rearranging gives:
\[ x = y - 3 \]

### Step 5: Write the inverse function
Thus, the inverse function is:
\[ f^{-1}(y) = y - 3 \]

### Step 6: Determine the range for the inverse function
Since the range of \( f(x) \) is \( (9, 12) \), the domain of \( f^{-1}(y) \) is \( (9, 12) \).

### Final Result
The inverse function is:
\[ f^{-1}(x) = x - 3 \quad \text{for } x \in (9, 12) \]