Let `f : [2, 4) rarr [1, 3)` be a function defined by `f(x) = x - [(x)/(2)]` (where [.] denotes the greatest integer function). Then `f^(-1)` equals

Options

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

Answer

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

### Step 1: Understand the function
The function \( f(x) = x - \left\lfloor \frac{x}{2} \right\rfloor \) involves the greatest integer function, which means we need to evaluate how \( \left\lfloor \frac{x}{2} \right\rfloor \) behaves in the interval \( [2, 4) \).

### Step 2: Evaluate \( \left\lfloor \frac{x}{2} \right\rfloor \) for the domain
- For \( x \) in the interval \( [2, 4) \):
- If \( x = 2 \), then \( \frac{2}{2} = 1 \) and \( \left\lfloor 1 \right\rfloor = 1 \).
- If \( x = 3 \), then \( \frac{3}{2} = 1.5 \) and \( \left\lfloor 1.5 \right\rfloor = 1 \).
- If \( x \) approaches \( 4 \) (but is less than \( 4 \)), \( \frac{4}{2} = 2 \) and \( \left\lfloor 2 \right\rfloor = 2 \).

Thus, for \( x \) in \( [2, 4) \):
- \( \left\lfloor \frac{x}{2} \right\rfloor = 1 \) for \( x \in [2, 4) \) except at \( x = 4 \) where it would be \( 2 \).

### Step 3: Simplify \( f(x) \)
Since \( \left\lfloor \frac{x}{2} \right\rfloor = 1 \) for \( x \in [2, 4) \):
\[ f(x) = x - 1 \]
This means that \( f(x) \) maps \( [2, 4) \) to \( [1, 3) \).

### Step 4: Find the inverse function
To find the inverse, we set \( y = f(x) \):
\[ y = x - 1 \]
Solving for \( x \):
\[ x = y + 1 \]
Thus, the inverse function is:
\[ f^{-1}(y) = y + 1 \]

### Step 5: Replace \( y \) with \( x \) in the inverse function
To express the inverse function in standard form:
\[ f^{-1}(x) = x + 1 \]

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