If `f(x)=sin{[x+5]+{x-{x-{x}}}" for "x in (0,pi/4)` is invertible, where {.} and [.] represent fractional part and greatest integer functions respectively, then `f^(-1) (x)` is ::
`I. sin^(-1)x` II. `pi/2-cos^(-1)x` III. `sin^(-1){x}` IV`.cos^(-1){x}`
The correct choice is:

Options

A. I,II,III
B. II,III
C. III,IV
D. None of these

Answer

To solve the problem, we need to analyze the function \( f(x) = \sin\{[x+5] + \{x - \{x - \{x\}\}\}\} \) for \( x \) in the interval \( (0, \frac{\pi}{4}) \) and determine its inverse.

### Step 1: Understanding the Components of the Function
1. **Greatest Integer Function**: The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \).
2. **Fractional Part Function**: The fractional part function \( \{x\} \) is defined as \( x - [x] \).

### Step 2: Simplifying the Function
1. **Evaluate \( [x + 5] \)**:
- Since \( x \) is in the interval \( (0, \frac{\pi}{4}) \), \( x + 5 \) will be in the interval \( (5, 5 + \frac{\pi}{4}) \).
- Thus, \( [x + 5] = 5 \) (as it is the largest integer less than or equal to \( x + 5 \)).

2. **Evaluate \( \{x - \{x - \{x\}\}\} \)**:
- Since \( x \) is in \( (0, 1) \), \( \{x\} = x \).
- Therefore, \( \{x - \{x - \{x\}\}\} = \{x - \{x - x\}\} = \{x - 0\} = \{x\} = x \).

3. **Combine the Results**:
- Now we can substitute back into the function:
\[ f(x) = \sin\{5 + x\} \]

### Step 3: Finding the Inverse Function
1. **Determine the Range of \( f(x) \)**:
- As \( x \) varies from \( 0 \) to \( \frac{\pi}{4} \), \( 5 + x \) varies from \( 5 \) to \( 5 + \frac{\pi}{4} \).
- Therefore, \( f(x) = \sin(5 + x) \) will take values from \( \sin(5) \) to \( \sin(5 + \frac{\pi}{4}) \).

2. **Finding the Inverse**:
- The inverse function \( f^{-1}(y) \) can be expressed as:
\[ f^{-1}(y) = \sin^{-1}(y) \quad \text{or} \quad f^{-1}(y) = \frac{\pi}{2} - \cos^{-1}(y) \]
- Since \( \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2} \), both expressions are valid.

### Step 4: Conclusion
From the analysis, we find that the inverse function \( f^{-1}(x) \) can be expressed as:
- \( \sin^{-1}(x) \)
- \( \frac{\pi}{2} - \cos^{-1}(x) \)

Thus, the correct choices for \( f^{-1}(x) \) are:
- I. \( \sin^{-1} x \)
- II. \( \frac{\pi}{2} - \cos^{-1} x \)

### Final Answer
The correct choices are I and II.