Let `f :(4,6) → (6,8)` be a function defined by `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 - 2 `
C. `x - 2 `
D. `(1)/(x + [(x)/(2)]) `

Explanation

To find the inverse of the function \( f(x) = x + \left\lfloor \frac{x}{2} \right\rfloor \) defined on the interval \( (4, 6) \) and mapping to \( (6, 8) \), we will follow these steps:

### Step 1: Determine the range of \( \left\lfloor \frac{x}{2} \right\rfloor \)
Since \( x \) is in the interval \( (4, 6) \):
- The minimum value of \( \frac{x}{2} \) when \( x = 4 \) is \( \frac{4}{2} = 2 \).
- The maximum value of \( \frac{x}{2} \) when \( x = 6 \) is \( \frac{6}{2} = 3 \).

Thus, \( \frac{x}{2} \) ranges from \( 2 \) to \( 3 \). Therefore, the greatest integer function \( \left\lfloor \frac{x}{2} \right\rfloor \) will take the value \( 2 \) for all \( x \) in \( (4, 6) \).

### Step 2: Simplify the function
Since \( \left\lfloor \frac{x}{2} \right\rfloor = 2 \) for \( x \in (4, 6) \), we can simplify the function:
\[ f(x) = x + 2 \]

### Step 3: Set up the equation for the inverse
To find the inverse function \( f^{-1}(x) \), we start with:
\[ y = f(x) = x + 2 \]
Now, we need to express \( x \) in terms of \( y \):
\[ y = x + 2 \implies x = y - 2 \]

### Step 4: Write the inverse function
Thus, we can express the inverse function as:
\[ f^{-1}(x) = x - 2 \]

### Conclusion
The inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = x - 2 \]