Composite Of Functions

Concept: For composition , the output of must lie in the domain of .
• Domain of :
• Domain of :

Step 1: Apply composition condition.


Step 2: Convert into exponential form.


Step 3: Combine with domain of .

Step 1: Use the formula for arithmetic mean and geometric mean.
The arithmetic mean (AM) and geometric mean (GM) of two numbers and are given by: We are given that the ratio of AM to GM is , i.e.:
Step 2: Simplify the equation.
Simplifying the equation: Multiplying both sides by :
Step 3: Square both sides to eliminate the square root.
Squaring both sides of the equation: This simplifies to:
Step 4: Express .
Now, we need to find . From the above equation: Thus,

Concept:
• Composition of functions:
• Evaluate step-by-step instead of expanding fully

Step 1: Find


Step 2: Find


Step 3: Find again
Final Conclusion: