Find the inverse of the function: `f:(2,3) to (0,1)` defined by `f(x)=x-[x],` where[.] represents the greatest integer function

Explanation

To find the inverse of the function \( f: (2,3) \to (0,1) \) defined by \( f(x) = x - [x] \), where \([.]\) represents the greatest integer function, we can follow these steps:

### Step 1: Understand the function
The function \( f(x) = x - [x] \) gives the fractional part of \( x \). For \( x \) in the interval \( (2, 3) \), we have:
- \( [x] = 2 \) (since the greatest integer less than \( x \) in this interval is 2).

Thus, we can rewrite the function as:
\[ f(x) = x - 2 \]

### Step 2: Set up the equation for the inverse
To find the inverse, we set \( y = f(x) \):
\[ y = x - 2 \]

### Step 3: Solve for \( x \)
Now, we solve for \( x \) in terms of \( y \):
\[ y + 2 = x \]

### Step 4: Write the inverse function
Thus, the inverse function \( f^{-1}(y) \) is given by:
\[ f^{-1}(y) = y + 2 \]

### Step 5: Determine the range of the inverse function
Since \( f(x) \) maps \( (2, 3) \) to \( (0, 1) \), the inverse function will map \( (0, 1) \) back to \( (2, 3) \). Therefore, the inverse function is:
\[ f^{-1}: (0, 1) \to (2, 3) \]

### Final Answer
The inverse of the function \( f(x) = x - [x] \) is:
\[ f^{-1}(y) = y + 2 \]

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