Composite Of Functions

Step 1: Define the composition of functions.
We are given two functions:
- , which gives the absolute value of the cosine of ,
- , which is the greatest integer less than or equal to , i.e., the floor function.
We are asked to find , which is defined as , i.e., for .

Step 2: Apply .
First, we apply : Thus, gives the absolute value of , which is always non-negative.

Step 3: Apply .
Next, apply to . We have: Since is always between 0 and 1 for all values of , the greatest integer less than or equal to is 0 for all where .

Step 4: Conclusion.
Thus, , which is equivalent to option (C), .

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
Circle: . Line:
Intersection:
Points: and
Area of sector =
Area of triangle =
Area =
Step 3: Final Answer

Concept: Composition of function:

Step 1: Substitute into itself.


Step 2: Apply definition of .


Step 3: Simplify denominator.


Step 4: Final simplification.
Conclusion:

Concept: Composition of functions:

Step 1: Assume from options.
Try

Step 2: Check .


Step 3: Check .


Step 4: Use identity.
Conclusion:
Correct functions = Option (B)