1. Lesson 1.5 Combinations of Functions
Essential Question: How do you combine two functions to form a new function?

2. Before we start… Given the following two expressions: 3𝑥−2 and 9𝑥+1
Perform the following operations:
Add
Subtract
Multiply
Divide

3. Arithmetic Combinations of Functions
Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to create new functions.

4. 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 follows.
Sum: 𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔 𝑥
Difference: 𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔 𝑥
Product: 𝑓𝑔 𝑥 =𝑓 𝑥 ∙𝑔 𝑥
Quotient: 𝑓 𝑔 𝑥 = 𝑓 𝑥 𝑔 𝑥 , 𝑔 𝑥 ≠0

5. Given 𝑓 𝑥 = 𝑥 2 and 𝑔 𝑥 =1−𝑥, find 𝑓+𝑔 𝑥 .

6. Given 𝑓 𝑥 = 𝑥 2 and 𝑔 𝑥 =1−𝑥, find 𝑓−𝑔 𝑥 .

7. Find 𝑔𝑓 𝑥 given that 𝑓 𝑥 = 1 𝑥 and 𝑔 𝑥 = 𝑥 𝑥+1
Find 𝑔𝑓 𝑥 given that 𝑓 𝑥 = 1 𝑥 and 𝑔 𝑥 = 𝑥 𝑥+1 . Then evaluate the product when 𝑥=−3.

8. Given 𝑓 𝑥 = 𝑥 2 and 𝑔 𝑥 =1−𝑥, find 𝑓 𝑔 𝑥 . Then find the domain of 𝑓 𝑔 𝑥 .

9. What is the composition of functions?
The composition of the function f with the function g is 𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥 .
The domain of 𝑓∘𝑔 is the set if all x in the domain of g such that 𝑔 𝑥 is in the domain of f.

10. Combining two functions into one new composite function.
With composition of functions, you start with the inside and work your way to the outside.
It’s evaluating a function but replacing x with another function instead of a number.

11. Composite Function

12. Find 𝑓∘𝑔 𝑥 for 𝑓 𝑥 = 1 𝑥 and 𝑔 𝑥 =2𝑥+1
Find 𝑓∘𝑔 𝑥 for 𝑓 𝑥 = 1 𝑥 and 𝑔 𝑥 =2𝑥+1. If possible, find 𝑓∘𝑔 −1 and 𝑓∘𝑔 −

13. Given 𝑓 𝑥 =2𝑥+5 and 𝑔 𝑥 =4 𝑥 2 +1, find the following.
𝑓∘𝑔 𝑥

14. Given 𝑓 𝑥 =2𝑥+5 and 𝑔 𝑥 =4 𝑥 2 +1, find the following.
𝑔∘𝑓 𝑥

15. Given 𝑓 𝑥 =2𝑥+5 and 𝑔 𝑥 =4 𝑥 2 +1, find the following.
𝑓∘𝑔 − 1 2

16. Find the composition 𝑓∘𝑔 𝑥 for the functions 𝑓 𝑥 = 𝑥 and 𝑔 𝑥 = 𝑥 2 +4
Find the composition 𝑓∘𝑔 𝑥 for the functions 𝑓 𝑥 = 𝑥 and 𝑔 𝑥 = 𝑥 Then find the domain of 𝑓∘𝑔 𝑥 .

17. Given 𝑓 𝑥 =3𝑥−2 and 𝑔 𝑥 = 1 3 𝑥+ 2 3 , find each composition.
𝑓∘𝑔 𝑥

18. Given 𝑓 𝑥 =3𝑥−2 and 𝑔 𝑥 = 1 3 𝑥+ 2 3 , find each composition.
𝑔∘𝑓 𝑥

19. Given 𝑓 𝑥 =𝑥−7 and 𝑔 𝑥 = 𝑥 2 , find a) 𝑓∘𝑔 b) 𝑔∘𝑓

20. Given 𝑓 𝑥 =𝑥+5 and 𝑔 𝑥 = 𝑥 2 −3, find a) 𝑓∘𝑔 b) 𝑔∘𝑓

21. Given 𝑓 𝑥 =𝑥−1 and 𝑔 𝑥 = 1 𝑥 2 , find a) 𝑓∘𝑔 b) 𝑔∘𝑓

22. Write the function ℎ 𝑥 = 1 𝑥 2 −2 as a composition of two functions.

23. Write the function ℎ 𝑥 = 1 𝑥−2 2 as a composition of two functions.

24. Write the function ℎ 𝑥 = 1 𝑥+7 as a composition of two functions.

25. The weekly cost of producing x units in a manufacturing process is given by the function 𝐶 𝑥 =48𝑥 The number of units produced in t hours is given by 𝑥 𝑡 =40𝑡. Find and interpret 𝐶∘𝑥 𝑡 .

26. The number N of bacteria in a refrigerated petri dish is given by
𝑁(𝑇) = 20𝑇2 – 80𝑇 + 500, 2  𝑇  14
where T is the temperature of the petri dish (in degrees Celsius). When the petri dish is removed from refrigeration, the temperature of the petri dish is given by
𝑇 (𝑡) = 4𝑡 + 2, 0  𝑡  3
where t is the time (in hours).
a. Find the composition 𝑁 (𝑇(𝑡)) and interpret its meaning in context.
b. Find the number of bacteria in the petri dish when 𝑡 = hours.
c. Find the time when the bacteria count reaches 2000.

27. How do you combine two functions to form a new function?

28. Ticket Out the Door
Given 𝑓 𝑥 = 𝑥 2 and 𝑔 𝑥 =3𝑥+2, find 𝑓∘𝑔