1. Inverses of Functions
Section 1.6

2. Objective
To be able to find inverses of functions.

3. Relevance
To be able to model a set of raw data after a function to best represent that data.

4. Go Over Quiz

5. Warm Up: Graph & give the domain & range.
Answer on Next Slide

6. Warm Up #3: Graph & give the domain & range.x y x y -3 -4 -5 2 1 1 2 3 4 -3 -2 -1

7. Intro to Inverses Sneetches Video or Read Story Quickly
A function describes the relationship between 2 variables, applying a rule to an input that generates exactly one output.
For such relationships, we are often compelled to “reverse” or “undo” the rule.

8. The Sneetches Story
Some of the Sneetches have stars on their yellow bellies; some do not.
In the story, Sylvester McMonkey McBean builds a machine that can apply these stars.
Sneetches without stars pay McBean to make them look like their start-bellied friends.
When all Sneetches appear the same, another problem emerges. So………………

9. The Sneetches Story
McBean builds a second machine to “reverse” the action of the first one – one that removes the stars.
Can you identify the math connections to this story?

10. Math Connections functions McBean’s machine represents____________.
The second machine “reverses” the action of the first, thereby acting as an ____________.
inverse

11. Progress of Inverses Throughout Math
Learned Addition and then its inverse operation Subtraction.
Learned Multiplication and then its inverse operation Division.
Learning Perfect Squares connects with extracting Square Roots
Basically – Inverses are a second operation that reverses the first one!

12. Calculator Inverses Take a look at your GDC’s and observe the keys.
Do you notice that inverse operations of many calculator commands are “second” functions?
A calculator key pairs an operation or function with its inverse.

13. Inverse Function Partner Share.
List the sequence of steps needed to evaluate this function:
1. Square the Input
2. Multiply by 9
3. Add 4

14. Inverse Functions Partner Share
Divide into groups of 3 or 4.
Each group will be given two functions.
For each function:
1) List the steps of each function
2) Provide 3 ordered pairs that will
satisfy the function.
Seek out other teams who had the other half of the function share information. For example, a team that had 2A will seek out a team that had 2B.
Class Discussion – What do you notice about the “undoing” of inverses of functions?

15. Warm Up #3: Graph & give the domain & range.
Answer on Next Slide

16. Warm Up #3: Graph & give the domain & range.x y x y -3 -4 -5 2 1 1 2 3 4 -3 -2 -1

17. Inverse of a relation
The inverse of the ordered pairs (x, y) is the set of all ordered pairs (y, x).
The Domain of the function is the range of the inverse and the Range of the function is the Domain of the inverse.
Symbol:
In other words, switch the
x’s and y’s!

18. Example: {(1,2), (2, 4), (3, 6), (4, 8)}
Inverse:

19. Function notation? What is really happening when you find the inverse?
Find the inverse of f(x)=4x-2 *4 -2 x
4x-2
(x+2)/4 /4 +2 x So

20. To find an inverse… Switch the x’s and y’s. Solve for y.
Change to functional notation.

21. Find Inverse:

22. Find Inverse:

23. Find Inverse:

24. Find Inverse:

25. Draw the inverse. Compare to the line y = x. What do you notice?

26. Graph the inverse of the following:
The function and
its inverse are
symmetric with
respect to the
Line y = x. x y -3 1 -5 -4 2 4

27. How Does an Inverse Exist

28. More on One-To-One
Recall that a function is a set of ordered pairs where every first coordinate has exactly one second coordinate.
A one-to-one function has the added constraint that each 2nd coordinate has exactly one 1st coordinate.

29. Example

30. Example

31. Example

32. Things to note.. The domain of is the range of f(x).
The graph of an inverse function can be found by reflecting a function in the line y=x.
Check this by plotting y = 3x + 1 and
on your graphic calculator.
Take a look

33. Reflecting..

34. Find the inverse of the function.
Is the inverse also a function? Let’s look at the graphs.
NOTE: Inverse is NOT
a function!
Inverse

35. Horizontal Line Test
Recall that a function passes the vertical line test.
The graph of a one-to-one function will pass the horizontal line test. (A horizontal line passes through the function in only one place at a time.)

36. Is it an Inverse?
A function can only have an inverse if it is one-to-one.
You can use the horizontal line test on graphical representations to see if the function is one-to-one.

37. Composition and Inverses
If f and g are functions and
then f and g are inverses of one another.
!!!!!!!!!!!!!!!!!!!!!!

38. Example: Show that the following are inverses of each other.
The composition of each both produce
a value of x; Therefore, they are inverses
of each other.

39. Are f & g inverses?

40. You Try….
Show that
are inverses of each other.

41. Are f & g inverses?