1. Properties Of Functions 1.3

2. An even function is a function that is symmetric to the y-axis.
Even Functions
A function f is even if
An even function is a function that is symmetric to the y-axis.

3. Odd Functions
A function f is odd if
In other words an odd function is a function that is symmetric to the origin.
A function does not have to be even or odd it may be neither

4. Determine if the graph is even, odd or neither.
Even Function
Symmetric to y-axis
Neither
Even Function
Symmetric to y-axis
Odd
Function
Symmetric to origin

5. Determine if the graph is even, odd or neither.
Function
Symmetric to origin
Odd
Function
Symmetric to origin
Even Function
Symmetric to y-axis
Neither

6. To show if an equation is even or odd 1st find f(-x) and –f(x)
If you are asked to show or prove they are even or odd, you must show f(-x) and –f(x).
A function cannot be even and odd so if it passes one test you can conclude the other test will be false.
If this is true the function is even
If this is true the function is odd
What this shows is if
What this shows is if
then
then

7. To find –f(x), multiply the equation by -1

8. Remember this slide?
positive
negative
negative
positive

10. Neither To show if an equation is even or odd 1st find f(-x) and –f(x)
If you are asked to show they are even or odd you must show work not just look at the graph
Neither
The function is not even
The function is not odd

11. To show if an equation is even or odd 1st find f(-x) and –f(x)
If you are asked to show they are even or odd you must show work not just look at the graph
Even Function
The function is even
The function is not odd

12. Show whether the following functions are even or odd
Neither

13. Show whether the following functions are even or odd
The function is odd

14. To show if an equation is even or odd 1st find f(-x) and –f(x)
If you are asked to show they are even or odd you must show work not just look at the graph
Odd function
The function is not even
The function is odd

15. P odd, 29-32

16. Increasing
To describe an interval where a graph is increasing, give the x values where the graph is going up as a point moves to the right on the graph.

17. The function is increasing from (-7,1) or (6,9)
The interval will never include the endpoints

18. Decreasing
To describe an interval where a graph is decreasing, give the x values where the graph is going down as a point moves to the right on the graph.

19. The function is decreasing fromOr
The interval will never include the endpoints

20. To describe where a graph is constant, give the x values where the graph is horizontal.
The interval will never include the endpoints

21. The graph is constant fromor

22. relative maxima

23. There is a relative maximum at x=1. The relative maximum value is 8
At x =9 there is not considered to be a relative maximum

24. Relative Minima

25. There are relative minima at x=-7 and x=6.
f(-2) = 6, f(-7) = -10
At x = 6 the relative minimum value is -2
At x = -7 the relative minimum value is -10

26. Give the numbers if any, at which f has a relative maximum
Give the numbers if any, at which f has a relative maximum. What are these relative maxima?
Give the numbers if any, at which f has a relative minimum. What are these relative minima?
c) Where the graph is increasing? d) Where is the graph decreasing?

27. Give the numbers if any, at which f has a relative maximum
Give the numbers if any, at which f has a relative maximum. What are these relative maxima? At x=0 there is a relative maximum. f(0)= 2.5
Give the numbers if any, at which f has a relative minimum. What are these relative minima? At x=3 there is a relative minimum. f(3)= -10
c) Where the graph is increasing? d) Where is the graph decreasing?

28. 4 4 x=-5.4, -2 or 7 no At x=-4 there is a relative minimum, f(-4 )=-4
Interval(s) on which f is increasing
Interval(s) on which f is decreasing
Interval(s) on which f is constant
Interval(s) on which f is positive
Interval(s) on which f is negative 4
f(1)= ______
f(0)=
Values of x for which f(x)=0
Is f(3) a relative maximum?
Any relative minima and the numbers where they occur 4
x=-5.4, -2 or 7 no
At x=-4 there is a relative minimum, f(-4 )=-4
-5.4

29. P odd, 33

30. P odd, 33

31. Find

32. Find
Put in homework

33. Find
1st find f(x+h)

34. 2nd plug in

35. Find
3rd simplify

36. Put in homework
1st find f(x+h)
Find

38. Two Problems from powerpoint
12, 14, 20, 24, 26, 34, 57, 61, 67

39. P 182 – , 14, 20, 24, 26, 34, 57, 61, 67

40. P 182 – , 14, 20, 24, 26, 34, 57, 61, 67

41. Do Piecewise Worksheet

42. Piecewise Functions +

43. A piecewise function is a function that has different rules for different parts of the domain
These are the subdomains that tells us when to use each rule
These are the three rules that will give us the value of the function

44. Start by graphing the first rule or function.

45. The subdomain for this part of the graph is x<-1
Erase the part of the graph that is not less than -1.
In other words put an open circle at -1 and erase to the right of -1

46. Next graph the second rule or equation

47. The subdomain of is . Erase any part of the graph that is outside of this subdomain.
In other words put a solid dot at -1 and erase to the left of it. Also, put a solid dot at 1 and erase to the right.

48. Next graph

49. The subdomain for the third rule is x>1
The subdomain for the third rule is x>1. Therefore, we only want the graph of to the right of x=1.
Normally we would put an open dot on the graph at x=1 but it overlaps the endpoint of the second graph which is already solid therefore, we just erase to the left of x=1.

50. Find the Range

51. Range
Find the following

52. Graph and find the range

53. Graph and find the range

54. P odd