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 function f (x) is

Options

A. periodic
B. even
C. odd
D. neither odd nor even

Answer

To solve the problem, we need to analyze the four 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 set \( S \)

The functions in the set \( S \) are:
1. \( [x] \) - Greatest integer function (floor function)
2. \( \sin^{-1} x \) - Inverse sine function
3. \( |x| \) - Modulus function
4. \( \{x\} \) - Fractional part function

### Step 2: Determine the properties of each function

1. **Odd Functions (Set \( A \))**:
- A function \( f(x) \) is odd if \( f(-x) = -f(x) \).
- \( \sin^{-1} x \) is odd: \( \sin^{-1}(-x) = -\sin^{-1}(x) \).
- \( [x] \) is neither odd nor even.
- \( |x| \) is even.
- \( \{x\} \) is neither odd nor even.

Thus, the only function in \( A \) is \( \sin^{-1} x \).

2. **Discontinuous Functions (Set \( B \))**:
- The greatest integer function \( [x] \) is discontinuous at every integer.
- The fractional part function \( \{x\} \) is also discontinuous at every integer.
- \( |x| \) and \( \sin^{-1} x \) are continuous everywhere.

Therefore, the functions in \( B \) are \( [x] \) and \( \{x\} \).

3. **Non-decreasing Functions (Set \( C \))**:
- A function is non-decreasing if \( f(x_1) \leq f(x_2) \) for \( x_1 < x_2 \).
- \( \sin^{-1} x \) is non-decreasing on its domain.
- \( |x| \) is non-decreasing.
- \( [x] \) is non-decreasing but has discontinuities.
- \( \{x\} \) is non-decreasing but has discontinuities.

Thus, the functions in \( C \) are \( \sin^{-1} x \), \( |x| \), and \( [x] \).

### Step 3: Identify the intersections

- \( A \cap C \): The only function that is both odd and non-decreasing is \( \sin^{-1} x \).
- \( B \cap C \): There are no functions that are both discontinuous and non-decreasing.
- \( h(x) \in B \) but not in \( C \): This can be \( [x] \) or \( \{x\} \).
- \( l(x) \) is neither in \( A \), \( B \), nor \( C \): This can be \( |x| \).

### Conclusion

Since \( f(x) \in A \cap C \), we conclude that:

**The function \( f(x) \) is \( \sin^{-1} x \)**.