2. Packet #12 Composite, One-to-one, and Inverse Functions
Math 160
Packet #12
Composite, One-to-one, and Inverse Functions

3. Another way to combine two functions 𝑓(𝑥) and 𝑔(𝑥) is to plug one into the other. This is called ____________________. The composite function 𝑓∘𝑔 is defined by 𝒇∘𝒈 𝒙 =𝒇 𝒈 𝒙 .

4. Another way to combine two functions 𝑓(𝑥) and 𝑔(𝑥) is to plug one into the other. This is called ____________________. The composite function 𝑓∘𝑔 is defined by 𝒇∘𝒈 𝒙 =𝒇 𝒈 𝒙 .
function composition

5. Another way to combine two functions 𝑓(𝑥) and 𝑔(𝑥) is to plug one into the other. This is called ____________________. The composite function 𝑓∘𝑔 is defined by 𝒇∘𝒈 𝒙 =𝒇 𝒈 𝒙 .
function composition

6. The domain of 𝑓∘𝑔 is the set of all ___ in the domain of ___ such that _____ is in the domain of ___. 𝑔 𝑓 𝒙

7. The domain of 𝑓∘𝑔 is the set of all ___ in the domain of ___ such that _____ is in the domain of ___. 𝑔 𝑓 𝒙
𝒈(𝒙)

8. The domain of 𝑓∘𝑔 is the set of all ___ in the domain of ___ such that _____ is in the domain of ___. 𝑔 𝑓 𝒙
𝒈(𝒙)
𝒇(𝒈 𝒙 )

9. The domain of 𝑓∘𝑔 is the set of all ___ in the domain of ___ such that _____ is in the domain of ___. 𝑔 𝑓 𝒙
𝒈(𝒙)
𝒇(𝒈 𝒙 )

10. The domain of 𝑓∘𝑔 is the set of all ___ in the domain of ___ such that _____ is in the domain of ___. 𝑔 𝑓 𝒙
𝒈(𝒙)
𝒇(𝒈 𝒙 ) 𝒙

11. The domain of 𝑓∘𝑔 is the set of all ___ in the domain of ___ such that _____ is in the domain of ___. 𝑔 𝑓 𝒙
𝒈(𝒙)
𝒇(𝒈 𝒙 ) 𝒙 𝒈

12. The domain of 𝑓∘𝑔 is the set of all ___ in the domain of ___ such that _____ is in the domain of ___. 𝑔 𝑓 𝒙
𝒈(𝒙)
𝒇(𝒈 𝒙 ) 𝒙 𝒈
𝒈(𝒙)

13. The domain of 𝑓∘𝑔 is the set of all ___ in the domain of ___ such that _____ is in the domain of ___. 𝑔 𝑓 𝒙
𝒈(𝒙)
𝒇(𝒈 𝒙 ) 𝒙 𝒈
𝒈(𝒙) 𝒇

14. Ex 1. Let 𝑓 𝑥 = 𝑥 2 and 𝑔 𝑥 =𝑥−3. Find the functions 𝑓∘𝑔 and 𝑔∘𝑓
Ex 1. Let 𝑓 𝑥 = 𝑥 2 and 𝑔 𝑥 =𝑥−3. Find the functions 𝑓∘𝑔 and 𝑔∘𝑓. Also find their domains. 𝑓∘𝑔 5 = 𝑔∘𝑓 7 =

15. Ex 1. Let 𝑓 𝑥 = 𝑥 2 and 𝑔 𝑥 =𝑥−3. Find the functions 𝑓∘𝑔 and 𝑔∘𝑓
Ex 1. Let 𝑓 𝑥 = 𝑥 2 and 𝑔 𝑥 =𝑥−3. Find the functions 𝑓∘𝑔 and 𝑔∘𝑓. Also find their domains. 𝑓∘𝑔 5 = 𝑔∘𝑓 7 =

16. Ex 2. Let 𝑓 𝑥 = 𝑥 and 𝑔 𝑥 = 2−𝑥. Find the functions 𝑓∘𝑔 and 𝑔∘𝑓
Ex 2. Let 𝑓 𝑥 = 𝑥 and 𝑔 𝑥 = 2−𝑥 . Find the functions 𝑓∘𝑔 and 𝑔∘𝑓. Also find their domains.

17. A function is ____________ if 𝑓 𝑥 1 ≠𝑓( 𝑥 2 ) whenever 𝑥 1 ≠ 𝑥 2
A function is ____________ if 𝑓 𝑥 1 ≠𝑓( 𝑥 2 ) whenever 𝑥 1 ≠ 𝑥 2 . In other words, a function is one-to-one if two different inputs always give you two different outputs.

18. A function is ____________ if 𝑓 𝑥 1 ≠𝑓( 𝑥 2 ) whenever 𝑥 1 ≠ 𝑥 2
A function is ____________ if 𝑓 𝑥 1 ≠𝑓( 𝑥 2 ) whenever 𝑥 1 ≠ 𝑥 2 . In other words, a function is one-to-one if two different inputs always give you two different outputs.
one-to-one

19. A function is ____________ if 𝑓 𝑥 1 ≠𝑓( 𝑥 2 ) whenever 𝑥 1 ≠ 𝑥 2
A function is ____________ if 𝑓 𝑥 1 ≠𝑓( 𝑥 2 ) whenever 𝑥 1 ≠ 𝑥 2 . In other words, a function is one-to-one if two different inputs always give you two different outputs.
one-to-one

20. We can test this visually by using the __________________, which says that a function is one-to-one if no horizontal line intersects its graph more than once.

21. We can test this visually by using the __________________, which says that a function is one-to-one if no horizontal line intersects its graph more than once.
Horizontal Line Test

28. Inverse Functions
Let 𝑓 be a one-to-one function with domain A and range B. The _______ of 𝑓, called 𝑓 −1 , has domain B and range A, and is defined by 𝑓 −1 𝑦 =𝑥 ⇔ 𝑓 𝑥 =𝑦 for any 𝑦 in B. So, any input/output pair of 𝑓 is switched for 𝑓 −1 .

29. Inverse Functions
Let 𝑓 be a one-to-one function with domain A and range B. The _______ of 𝑓, called 𝑓 −1 , has domain B and range A, and is defined by 𝑓 −1 𝑦 =𝑥 ⇔ 𝑓 𝑥 =𝑦 for any 𝑦 in B. So, any input/output pair of 𝑓 is switched for 𝑓 −1 .
inverse

30. Inverse Functions
Let 𝑓 be a one-to-one function with domain A and range B. The _______ of 𝑓, called 𝑓 −1 , has domain B and range A, and is defined by 𝑓 −1 𝑦 =𝑥 ⇔ 𝑓 𝑥 =𝑦 for any 𝑦 in B. So, any input/output pair of 𝑓 is switched for 𝑓 −1 .
inverse

31. Inverse functions “undo” other functions
Inverse functions “undo” other functions. For example, if 𝑓 𝑥 =2𝑥+1, then 𝑓 −1 𝑥 = 𝑥−1 2 . To show that 2𝑥+1 and 𝑥−1 2 are inverse functions, we can use the following properties: 𝑓 −1 𝑓 𝑥 = ___ for every 𝑥 in A 𝑓 𝑓 −1 𝑥 = ___ for every 𝑥 in B

32. Inverse functions “undo” other functions
Inverse functions “undo” other functions. For example, if 𝑓 𝑥 =2𝑥+1, then 𝑓 −1 𝑥 = 𝑥−1 2 . To show that 2𝑥+1 and 𝑥−1 2 are inverse functions, we can use the following properties: 𝑓 −1 𝑓 𝑥 = ___ for every 𝑥 in A 𝑓 𝑓 −1 𝑥 = ___ for every 𝑥 in B

33. Inverse functions “undo” other functions
Inverse functions “undo” other functions. For example, if 𝑓 𝑥 =2𝑥+1, then 𝑓 −1 𝑥 = 𝑥−1 2 . To show that 2𝑥+1 and 𝑥−1 2 are inverse functions, we can use the following properties: 𝑓 −1 𝑓 𝑥 = ___ for every 𝑥 in A 𝑓 𝑓 −1 𝑥 = ___ for every 𝑥 in B 𝒙

34. Inverse functions “undo” other functions
Inverse functions “undo” other functions. For example, if 𝑓 𝑥 =2𝑥+1, then 𝑓 −1 𝑥 = 𝑥−1 2 . To show that 2𝑥+1 and 𝑥−1 2 are inverse functions, we can use the following properties: 𝑓 −1 𝑓 𝑥 = ___ for every 𝑥 in A 𝑓 𝑓 −1 𝑥 = ___ for every 𝑥 in B 𝒙 𝒙

36. Ex 3. Show that 𝑓 𝑥 =2𝑥+1 and 𝑔 𝑥 = 𝑥−1 2 are inverses of each other.

37. Note: Only one-to-one functions have inverses
Note: Only one-to-one functions have inverses. (Also, one-to-one functions always have inverses.)

38. Note: Only one-to-one functions have inverses
Note: Only one-to-one functions have inverses. (Also, one-to-one functions always have inverses.)

39. How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2
How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2. __________ 𝑥 and 𝑦. 3. Solve for ____. 4. Replace 𝑦 with _________.

40. How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2
How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2. __________ 𝑥 and 𝑦. 3. Solve for ____. 4. Replace 𝑦 with _________. 𝒚

41. How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2
How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2. __________ 𝑥 and 𝑦. 3. Solve for ____. 4. Replace 𝑦 with _________. 𝒚
Switch

42. How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2
How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2. __________ 𝑥 and 𝑦. 3. Solve for ____. 4. Replace 𝑦 with _________. 𝒚
Switch 𝒚

43. How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2
How to find the inverse of 𝒇(𝒙): 1. Replace 𝑓(𝑥) with ___. 2. __________ 𝑥 and 𝑦. 3. Solve for ____. 4. Replace 𝑦 with _________. 𝒚
Switch 𝒚
𝒇 −𝟏 (𝒙)

44. Ex 4. Find the inverse of 𝑓 𝑥 = 2𝑥+3 𝑥−1 .

45. Graphs and Inverses
Given any point (𝑎,𝑏) on the graph of 𝑓(𝑥), we can get a point on the graph of 𝑓 −1 (𝑥) by switching the coordinates: (𝑏,𝑎).

46. Graphs and Inverses
Ex 5. Given the graph of 𝑓(𝑥), draw the graph of 𝑓 −1 (𝑥).

47. Graphs and Inverses
Ex 5. Given the graph of 𝑓(𝑥), draw the graph of 𝑓 −1 (𝑥).

48. Graphs and Inverses
Ex 5. Given the graph of 𝑓(𝑥), draw the graph of 𝑓 −1 (𝑥).

49. Graphs and Inverses
Ex 5. Given the graph of 𝑓(𝑥), draw the graph of 𝑓 −1 (𝑥).

50. Graphs and Inverses
Ex 5. Given the graph of 𝑓(𝑥), draw the graph of 𝑓 −1 (𝑥).

51. Graphs and Inverses
Ex 5. Given the graph of 𝑓(𝑥), draw the graph of 𝑓 −1 (𝑥).

52. Graphs and Inverses
Notice that the entire graph of 𝑓 −1 𝑥 will be the mirror image of 𝑓 𝑥 across the 𝑦=𝑥 line.