If the function `f:R rarrR` be such that `f(x)=x-[x]`, where `[x]` denotes the greatest integer less than or equal to k, then `f^(-1)(x)` is :

Options

A. a. `[x]-x`
B. b. `(1)/(x-[x])`
C. c. not defined
D. d. none of these

Answer

To solve the problem of finding the inverse of the function \( f(x) = x - [x] \), where \([x]\) denotes the greatest integer less than or equal to \( x \), we can follow these steps:

### Step 1: Understand the function \( f(x) \)
The function \( f(x) = x - [x] \) represents the fractional part of \( x \). This means that for any real number \( x \), \( f(x) \) gives the part of \( x \) that is left after subtracting the greatest integer less than or equal to \( x \).

For example:
- If \( x = 2.4 \), then \( [x] = 2 \) and \( f(2.4) = 2.4 - 2 = 0.4 \).
- If \( x = 3.7 \), then \( [x] = 3 \) and \( f(3.7) = 3.7 - 3 = 0.7 \).
- If \( x = 5 \), then \( [x] = 5 \) and \( f(5) = 5 - 5 = 0 \).

### Step 2: Determine the range of \( f(x) \)
The output of \( f(x) \) is always in the interval \([0, 1)\) because it represents the fractional part of \( x \). Therefore, \( f(x) \) can take any value from 0 (inclusive) to 1 (exclusive).

### Step 3: Find the inverse function \( f^{-1}(y) \)
To find the inverse function, we need to express \( x \) in terms of \( y \). Since \( f(x) = y \) implies that \( y = x - [x] \), we can rearrange this to find \( x \):
\[ x = y + [x] \]
Now, since \( [x] \) is an integer, we can denote it as \( n \), where \( n = [x] \). Thus, we can write:
\[ x = y + n \]
where \( n \) is an integer that satisfies \( n \leq x < n + 1 \).

### Step 4: Express \( f^{-1}(y) \)
For a given \( y \) in the range \([0, 1)\), we can write:
\[ f^{-1}(y) = y + n \]
where \( n \) can be any integer. Therefore, the inverse function is not uniquely defined for each \( y \) because for each \( y \), there are infinitely many values of \( x \) corresponding to different integers \( n \).

### Conclusion
Thus, the inverse function \( f^{-1}(y) \) can be expressed as:
\[ f^{-1}(y) = y + n \quad \text{for any integer } n. \]