1. Functions: Notations and Definitions

2. An “ONTO” Function ONTONOT ONTO (Here: A=Domain, B=Range.)

3. A “ONE to ONE” Function One to one Not one to one. (Here: A=Domain, B=Range.)

4. How can we find the domains of functions?

5. How can we find the domains of functions? (continued)

6. Finding the Ranges of Functions

7. Find the Domains and Ranges of the following Functions.

8. More Notation

10. General Properties of Functions

11. Looking at Discontinuities

12. General Properties of Functions

13. Over Which Intervals are these Functions Increasing, Decreasing or Constant?

15. General Properties of Functions

16. Boundedness

17. General Properties of Functions

18. Local and Absolute Extrema

19. General Properties of Functions

20. Symmetry

21. General Properties of Functions

22. Asymptotes

23. Twelve Basic Functions See Figures 1.36 – 1.47 Pages 99 - 101

24. Piecewise Functions

25. Building Functions from Functions

26. Examples

27. Composition of Functions

28. Examples

29. More Examples

30. One More Example 3)A store offers a 15% discount on all items and a 20% discount to store employees. a)Write a model for the price found by taking off the 15% discount before the 20% discount. b)Write a model for the price found by taking off the 20% discount before the 15% discount. c)Which results in a cheaper price?

31. Defining Relations and Functions Implicitly

32. Defining Relations and Functions Parametrically

33. Another Example

34. Inverse Relations and Functions

35. Finding Inverse Functions

37. Modeling with Functions We can solve practical problems by modeling them with functions. 1)A parabolic satellite dish with maximum diameter of 24 inches and height of 6 inches is packaged with a cardboard cylinder lodges inside it for protective support. The diameter had a diameter of 12 inches. How high must it be to sit flush with the top of the dish?

38. More Modeling with Functions 2)Grain leaks through a hole in the bottom of a suspended storage bin at 8 cubic inches per minute. The leaking grain forms a cone whose height is always equal to its radius. If the height is 1 foot tall at 2:00 p.m., how tall will it be at 3:00 p.m.? 3)A car with tires that are 15 inches in radius moves at 70 miles per hour. How many rotations are made per second by the tires?

39. Graphical Transformations of Functions Two Types of Transformations:  Rigid: size and shape of graph are preserved. (Ex: translations, reflections, rotations)  Nonrigid: size and shape can change. (Scaling, vertical and horizontal stretching and shrinking.)

40. Translations

41. Reflections

42. Reflections of Even and Odd Functions What happens when we reflect even functions across the:  X-axis  Y-axis  Origin  Same question for odd functions.

43. Stretching or Shrinking Graphs (Scaling)

44. Combinations of Transformations