Q2. Use composition to decide if two functions are inverses of each other.

Background

Topic: Functions and Inverses

This question tests your understanding of function composition and how to verify if two functions are inverses. Two functions, and , are inverses if and only if and for all in their domains.

Key Terms and Formulas

• Function Composition:

• Inverse Function: is the function that "undoes" .

• Inverse Test: and

Step-by-Step Guidance (Part a)

1. Write down the given functions: and .

2. Set up the first composition: . Substitute into :

3. Simplify the numerator inside the fraction.

4. Set up the second composition: . Substitute into :

5. Simplify the expression inside the parentheses and then add 2.

Step-by-Step Guidance (Part b)

1. Write down the given functions: and .

2. Set up the first composition: . Substitute into :

3. Simplify the expression under the square root and then continue simplifying the composition.

4. Set up the second composition: . Substitute into :

5. Simplify the numerator and then the entire expression inside the parentheses before squaring and subtracting 3.

Try solving on your own before revealing the answer!