Math 160

1. Math 160 2.7 – One-to-One Functions and Their Inverses

2. A function is ____________ if whenever . In other words, a function is one-to-one if two different inputs always give you two different outputs.

3. A function is ____________ if whenever . In other words, a function is one-to-one if two different inputs always give you two different outputs. one-to-one

4. A function is ____________ if whenever . In other words, a function is one-to-one if two different inputs always give you two different outputs. one-to-one

5. 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.

6. 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

13. Inverse Functions Let be a one-to-one function with domain A and range B. The _______of , called , has domain B and range A, and is defined by for any in B. So, any input/output pair of is switched for .

14. Inverse Functions Let be a one-to-one function with domain A and range B. The _______of , called , has domain B and range A, and is defined by for any in B. So, any input/output pair of is switched for . inverse

15. Inverse Functions Let be a one-to-one function with domain A and range B. The _______of , called , has domain B and range A, and is defined by for any in B. So, any input/output pair of is switched for . inverse

16. Inverse functions “undo” other functions. For example, if , then . To show that and are inverse functions, we can use the following properties: ___ for every in A ___ for every in B

17. Inverse functions “undo” other functions. For example, if , then . To show that and are inverse functions, we can use the following properties: ___ for every in A ___ for every in B

18. Inverse functions “undo” other functions. For example, if , then . To show that and are inverse functions, we can use the following properties: ___ for every in A ___ for every in B

19. Inverse functions “undo” other functions. For example, if , then . To show that and are inverse functions, we can use the following properties: ___ for every in A ___ for every in B

21. Ex 1. Show that and are inverses of each other.

22. Note: Only one-to-one functions have inverses.

23. How to find the inverse of : 1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________.

24. How to find the inverse of : 1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________.

25. How to find the inverse of : 1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________. Switch

26. How to find the inverse of : 1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________. Switch

27. How to find the inverse of : 1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________. Switch

28. Ex 2.Find the inverse of .

29. Graphs and Inverses Ex 3.Given the graph of , draw the graph of .

30. Graphs and Inverses Ex 3.Given the graph of , draw the graph of .

31. Graphs and Inverses Ex 3.Given the graph of , draw the graph of .

32. Graphs and Inverses Ex 3.Given the graph of , draw the graph of .

33. Graphs and Inverses Ex 3.Given the graph of , draw the graph of .

34. Graphs and Inverses Given any point on the graph of , we can get a point on the graph of by switching the coordinates: . As a result, the entire graph of will be the mirror image of across the line.