1. Combinations of Functions
Composite Functions

2. Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as:
Sum
Difference
Product
Quotient

3. Example 1
Given and
Find the sum and difference of the functions.

4. Example 1
Given and
Find the sum and difference of the functions.

5. Example 1
Given and
Find the sum and difference of the functions.

6. Example 1
Given and
Find the product and quotient of the functions.

7. Example 1
Given and
Find the product of the functions.

8. Example 3 Given and Find the quotient of the functions.
The domain of g(x) is all numbers except 3, -1. Even though we canceled, this is still the domain.

9. Example 3 Given and Find the quotient of the functions.
The domain of f(x) is all numbers except -1. Even though we canceled, this is still the domain.

10. You Try
Given and
Find the sum and difference of the functions.

11. You Try
Given and
Find the sum and difference of the functions.

12. You Try
Given and
Find the product and quotient of the functions.

13. You Try Given and Find the product and quotient of the functions.
The domain is still

14. Composition of Functions
Definition of The Composition of Two Functions.
The composition of the function f with the function g is
The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.

15. Example 4 Given and , find the following. a)
This means go to the f function and replace x with the g function.

16. Example 4 Given and , find the following. a)
This means go to the f function and replace x with the g function.

17. Example 4 Given and , find the following. a)
This means go to the g function and replace x with the f function.

18. Example 4 Given and , find the following. b)
This means go to the f function and replace x with -2. Take that answer and put it into g.

19. Example 4 Given and , find the following. b)
You could also go to your answer for and replace x with a -2.

20. You Try
Given and , find the following. a) b) c) d)

21. You Try
Given and , find the following. a) b) c) d)

22. Finding the Domain of the Composition of Functions
Given and , find the composition f(g(x)). Then find the domain of f(g(x)).

23. Finding the Domain of the Composition of Functions
Given and , find the composition f(g(x)). Then find the domain of f(g(x)).
We take the f function and replace x with the g function.

24. Example 5
So the composition , and the domain of this function is all real numbers. However, this is not the domain of the composition. We need to look at the domain of g(x) to know the domain of the composition.
, so its domain is where

25. Example 5
So the composition , and the domain of this function is all real numbers. However, this is not the domain of the composition. We need to look at the domain of g(x) to know the domain of the composition.
, so its domain is where
We need to do sign analysis.
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26. Class work
Page 238
6, 8, 14, 16, 20, 22, 36, 42

27. Homework
Page 238
5-19 odd
35,37,43