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 \( f(x) = x + \left\lfloor \frac{x}{2} \right\rfloor \) defined on the interval \( (6, 8) \), we will follow these steps:

### Step 1: Determine the value of \( \left\lfloor \frac{x}{2} \right\rfloor \)
Given the domain \( 6 < x < 8 \), we can calculate \( \frac{x}{2} \):
- For \( x = 6 \), \( \frac{6}{2} = 3 \)
- For \( x = 8 \), \( \frac{8}{2} = 4 \)

Thus, \( \frac{x}{2} \) lies in the interval \( (3, 4) \). The greatest integer function \( \left\lfloor \frac{x}{2} \right\rfloor \) will therefore be:
\[ \left\lfloor \frac{x}{2} \right\rfloor = 3 \]
for all \( x \) in the interval \( (6, 8) \).

### Step 2: Rewrite the function \( f(x) \)
Substituting \( \left\lfloor \frac{x}{2} \right\rfloor \) into the function:
\[ f(x) = x + 3 \]

### Step 3: Find the range of \( f(x) \)
Now, we need to find the range of \( f(x) \) for \( x \) in \( (6, 8) \):
- When \( x = 6 \), \( f(6) = 6 + 3 = 9 \)
- When \( x = 8 \), \( f(8) = 8 + 3 = 11 \)

Thus, the range of \( f(x) \) is \( (9, 11) \).

### Step 4: Set up the equation for the inverse function
To find the inverse function \( f^{-1}(x) \), we set:
\[ y = f(x) = x + 3 \]
We need to express \( x \) in terms of \( y \):
\[ y = x + 3 \implies x = y - 3 \]

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

### Conclusion
The inverse function \( f^{-1}(x) \) is:
\[ \boxed{x - 3} \]