Let S denotes the set consisting of four functions and `S = { [x], sin^(-1) x, |x|,{x}}` where , `{x}` denotes fractional part and [x] denotes greatest integer function , Let A, B , C are subsets of S.
Suppose
A : consists of odd functions (s)
B : consists of discontinuous function (s)
and C: consists of non-decreasing function(s) or increasing function (s).
If `f(x) in A nn C, g(x) in B nnC, h (x) in B" but not C and " l(x) in ` neither A nor B nor C .
Then, answer the following.
The range of `f(h(x))` is

Options

A. `(0, pi/2)`
B. `[0, pi/2)`
C. `(0, pi/2]`
D. `[0, pi/2]`

Answer

To solve the problem, we need to analyze the functions given in the set \( S = \{ [x], \sin^{-1} x, |x|, \{x\} \} \) and categorize them into the subsets \( A \), \( B \), and \( C \) based on their properties.

### Step 1: Identify the Functions in Each Subset

1. **Odd Functions (Subset A)**:
- A function \( f(x) \) is odd if \( f(-x) = -f(x) \).
- Among the functions in \( S \):
- \( \sin^{-1} x \) is an odd function.
- \( [x] \) (greatest integer function) is not odd.
- \( |x| \) is not odd.
- \( \{x\} \) (fractional part function) is not odd.
- Therefore, \( A = \{ \sin^{-1} x \} \).

2. **Discontinuous Functions (Subset B)**:
- A function is discontinuous if it has jumps or breaks.
- Among the functions in \( S \):
- \( [x] \) is discontinuous.
- \( \{x\} \) is discontinuous.
- \( \sin^{-1} x \) is continuous.
- \( |x| \) is continuous.
- Therefore, \( B = \{ [x], \{x\} \} \).

3. **Non-decreasing Functions (Subset C)**:
- A function is non-decreasing if \( f(x_1) \leq f(x_2) \) for \( x_1 < x_2 \).
- Among the functions in \( S \):
- \( \sin^{-1} x \) is non-decreasing.
- \( |x| \) is non-decreasing.
- \( [x] \) is not non-decreasing.
- \( \{x\} \) is not non-decreasing.
- Therefore, \( C = \{ \sin^{-1} x, |x| \} \).

### Step 2: Determine the Functions in Each Intersection

- \( f(x) \in A \cap C \):
- Since \( A = \{ \sin^{-1} x \} \) and \( C = \{ \sin^{-1} x, |x| \} \), we have:
- \( f(x) = \sin^{-1} x \).

- \( g(x) \in B \cap C \):
- \( B = \{ [x], \{x\} \} \) and \( C = \{ \sin^{-1} x, |x| \} \), so there are no common functions.
- Therefore, \( g(x) \) cannot be determined from \( B \cap C \).

- \( h(x) \in B \) but not \( C \):
- Since \( B = \{ [x], \{x\} \} \) and neither of these functions is in \( C \), we can take \( h(x) = \{x\} \) (the fractional part function).

- \( l(x) \) is neither in \( A \), \( B \), nor \( C \):
- This could be \( |x| \) or any other function not listed in \( A \), \( B \), or \( C \).

### Step 3: Find the Range of \( f(h(x)) \)

Now we need to find \( f(h(x)) = f(\{x\}) = \sin^{-1}(\{x\}) \).

1. **Determine the Range of \( \{x\} \)**:
- The fractional part function \( \{x\} \) takes values in the interval \( [0, 1) \).

2. **Determine the Range of \( \sin^{-1}(y) \) for \( y \in [0, 1) \)**:
- The function \( \sin^{-1}(y) \) is defined for \( y \) in the interval \( [-1, 1] \).
- Therefore, the range of \( \sin^{-1}(y) \) when \( y \) is in \( [0, 1) \) is:
- From \( \sin^{-1}(0) = 0 \) to \( \sin^{-1}(1) = \frac{\pi}{2} \).
- However, since \( \{x\} \) never actually reaches 1, the upper limit is not included.

Thus, the range of \( f(h(x)) = \sin^{-1}(\{x\}) \) is \( [0, \frac{\pi}{2}) \).

### Final Answer
The range of \( f(h(x)) \) is \( [0, \frac{\pi}{2}) \).