Composite Of Functions

To solve the problem, we analyze the function under the given conditions and match each case in List-I to the correct properties in List-II.

1. Understanding the Functions and :
The function is defined as:



The function is defined as:

2. Analyzing for Each Case:
The function is given by:

.

Case (P):
Here, .

- For , and . Thus, .
- For or , and . Thus, .

The range of is , and is not one-one, not onto, and not differentiable everywhere (due to discontinuities at and ).

Match: (P) → (5).

Case (Q):
Here, .

- The function is differentiable everywhere, including at (as ).
- The range of is because oscillates between and , covering all real values as varies.

Thus, is differentiable on and onto.

Match: (Q) → (3).

Case (R):
Here, .

- For , .
- For or , .

The function is one-one because it is strictly increasing (its derivative is positive where defined). It is also onto because as , , and as , .

Match: (R) → (2).

Case (S):
Here, .

- The range of is because for , decreases linearly from 1 to 0, and otherwise.
- The function is not one-one, not onto , and not differentiable at and .

Match: (S) → (4).

Final Answer:
The correct matching is:
.