Q1. Find the composition of two functions and the domain of the composition.

Background

Topic: Function Composition and Domain

This question tests your understanding of how to combine two functions using composition (i.e., or ) and how to determine the domain of the resulting function.

Key Terms and Formulas

• Function Composition: The composition of and is written as .

• Domain of Composition: The domain of is all in the domain of such that is in the domain of .

Step-by-Step Guidance

1. Identify the two functions you are asked to compose. Let's call them and .

2. Write the composition or , depending on the order specified.

3. Substitute the expression for into (or vice versa) to form the composite function.

4. To find the domain, first determine the domain of the inner function (e.g., ).

5. Next, find all such that is in the domain of . This may require solving inequalities or checking for restrictions (like division by zero or square roots of negative numbers).

Try solving on your own before revealing the answer!

Q2. Decompose a function as a composition of two functions.

Background

Topic: Function Decomposition

This question tests your ability to express a given function as the composition of two simpler functions.

Key Terms and Formulas

• Function Decomposition: Given a function , find functions and such that .

Step-by-Step Guidance

1. Look at the structure of the given function . Try to identify an "inner" function and an "outer" function such that .

2. Think about how you can rewrite in terms of a function inside another function. For example, if , you might let and .

3. Define as the part of that is "inside" another operation, and as the operation applied to $g(x)$.

4. Check your decomposition by composing and to see if you get back the original .

Try solving on your own before revealing the answer!