Let f: $(4,6)\rightarrow(6,8)$ be a function defined by $f(x)=x+[\frac{x}{2}]$ where [.] denotes the greatest integer function, then $f^{-1}(x)$ is:

Question

Accepted Answer

To determine the inverse function $ f^{-1}(x) $ for the given function $ f(x) = x + \left[\frac{x}{2}\right] $, where $[.]$ denotes the greatest integer function, we need to find a function that, when applied to $ f(x) $, returns the original input $ x $.

### Explanation for Option 3: $ x - 2 $

1. **Understanding the Function $ f(x) $:**
   - The function $ f(x) = x + \left[\frac{x}{2}\right] $ is defined for $ x \in (4, 6) $.
   - The greatest integer function $\left[\frac{x}{2}\right]$ returns the largest integer less than or equal to $\frac{x}{2}$.

2. **Behavior of $ f(x) $ in the Interval $ (4, 6) $:**
   - For $ x \in (4, 5) $, $\frac{x}{2}$ is between 2 and 2.5, so $\left[\frac{x}{2}\right] = 2$.
   - For $ x \in (5, 6) $, $\frac{x}{2}$ is between 2.5 and 3, so $\left[\frac{x}{2}\right] = 2$.

3. **Expression of $ f(x) $:**
   - In both sub-intervals, $ f(x) = x + 2 $.

4. **Finding the Inverse $ f^{-1}(x) $:**
   - To find $ f^{-1}(x) $, solve $ y = x + 2 $ for $ x $.
   - Rearranging gives $ x = y - 2 $.

5. **Conclusion:**
   - Therefore, the inverse function is $ f^{-1}(x) = x - 2 $, which matches Option 3.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $ x - \left[\frac{x}{2}\right] $**
   - This option suggests subtracting the greatest integer part of $\frac{x}{2}$ from $ x $.
   - This does not correctly reverse the operation of adding $\left[\frac{x}{2}\right]$ to $ x $ in the original function.

2. **Option 2: $-x - 2$**
   - This option suggests a linear transformation that does not correspond to the operation needed to reverse $ f(x) = x + 2 $.
   - It would not yield the original $ x $ when applied to $ f(x) $.

3. **Option 4: $\frac{1}{x + \left[\frac{x}{2}\right]}$**
   - This option implies a reciprocal operation, which is not relevant to reversing the addition of $\left[\frac{x}{2}\right]$.
   - It does not provide a valid inverse for the linear transformation $ f(x) = x + 2 $.

In summary, Option 3, $ x - 2 $, is the correct inverse function for $ f(x) = x + \left[\frac{x}{2}\right] $ in the given domain, as it correctly reverses the operation performed by $ f(x) $.