Types Of Functions

Step 1: Concept
To find the inverse, set and solve for in terms of .
Step 2: Analysis
.
Step 3: Conclusion
Replacing with , we get .
Final Answer: (A)

Step 1: Analyze the given function.
We are given the function . The function returns the maximum value among the three components , , and 2 for any value of .

Step 2: Examine the behavior of each part of the function.
- is a decreasing linear function.
- is an increasing linear function.
- The constant 2 is obviously always greater than or equal to the two linear functions for certain ranges of .

Step 3: Break the function into intervals.
We now find which part of the function is greater than the others in different ranges of .
- When , dominates.
- When , the maximum is 2.
- When , is less than , so dominates.

Step 4: Write the final equivalent function.
Thus, the function can be written as:

Step 5: Conclusion.
The correct equivalent function corresponds to option (B).

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}



Step 3: Final Answer

Concept: Recognize hyperbolic cosine identity:

Step 1: Use identity.


Step 2: Apply to given function.


Step 3: Compare with given expression.
But since scaling differs due to base , final does not match standard options. Conclusion:
Answer = None of these

Concept:

Step 1: Compute both sides

Step 2: Compare denominators We compare:

Step 3: Use inequality Since: But using:

Step 4: Take square roots and reciprocals Taking reciprocals (positive quantities): Conclusion

Concept:

Step 1: Check injectivity: Not injective.

Step 2: Check surjectivity: For any ,