Step Functions

1. Section 3-9
Step Functions

2. Warm-up
Name the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
4. .7777 5. -101.1 6. 2+ 3

3. Warm-up
Name the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
2
4. .7777 5. -101.1 6. 2+ 3

4. Warm-up
Name the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
2 3
4. .7777 5. -101.1 6. 2+ 3

5. Warm-up
Name the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
2 3 4
4. .7777 5. -101.1 6. 2+ 3

6. Warm-up
Name the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
2 3 4
4. .7777 5. -101.1 6. 2+ 3
0

7. Warm-up
Name the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
2 3 4
4. .7777 5. -101.1 6. 2+ 3
0 -102

8. Warm-up
Name the greatest integer that is less than or equal to the
following:
1. 2.99 2. π 3. 24
2 3 4
4. .7777 5. -101.1 6. 2+ 3
0 -102 3

9. Greatest-Integer:

10. Greatest-Integer: ⎢x ⎥
⎣ ⎦ The greatest integer less than or
equal to x

11. Greatest-Integer: ⎢x ⎥
⎣ ⎦ The greatest integer less than or
equal to x
Step Function:

12. Greatest-Integer: ⎢x ⎥
⎣ ⎦ The greatest integer less than or
equal to x
Step Function: A graph that looks like a series of steps,
with each “step” being a horizontal line segment

13. Greatest-Integer: ⎢x ⎥
⎣ ⎦ The greatest integer less than or
equal to x
Step Function: A graph that looks like a series of steps,
with each “step” being a horizontal line segment
*It is a function, so it must pass the Vertical-Line Test*

14. Greatest-Integer: ⎢x ⎥
⎣ ⎦ The greatest integer less than or
equal to x
Step Function: A graph that looks like a series of steps,
with each “step” being a horizontal line segment
*It is a function, so it must pass the Vertical-Line Test*
*Each step will include one endpoint*

15. Example 1
Simplify.
a. ⎢ 4 ⎥
⎣ ⎦ b. ⎢ −7 2 5 ⎥
⎣ ⎦ c. ⎢3.2 ⎥
⎣ ⎦

16. Example 1
Simplify.
a. ⎢ 4 ⎥
⎣ ⎦ b. ⎢ −7 2 5 ⎥
⎣ ⎦ c. ⎢3.2 ⎥
⎣ ⎦
4

17. Example 1
Simplify.
a. ⎢ 4 ⎥
⎣ ⎦ b. ⎢ −7 2 5 ⎥
⎣ ⎦ c. ⎢3.2 ⎥
⎣ ⎦
4 -8

18. Example 1
Simplify.
a. ⎢ 4 ⎥
⎣ ⎦ b. ⎢ −7 2 5 ⎥
⎣ ⎦ c. ⎢3.2 ⎥
⎣ ⎦
4 -8 3

19. Greatest-Integer
Function

20. Greatest-Integer
Function
The function f where f (x ) = ⎢ x ⎥
⎣ ⎦
for all real numbers x.

21. Greatest-Integer
Function
The function f where f (x ) = ⎢ x ⎥
⎣ ⎦
for all real numbers x.
*Also known as the rounding-down function*

22. Greatest-Integer
Function
The function f where f (x ) = ⎢ x ⎥
⎣ ⎦
for all real numbers x.
*Also known as the rounding-down function*
...because we’re rounding down

23. Example 2
Graph f (x ) = ⎢ x ⎥ +1.
⎣ ⎦

24. Example 2
Graph f (x ) = ⎢ x ⎥ +1.
⎣ ⎦
1. Set up a table

25. Example 2
Graph f (x ) = ⎢ x ⎥ +1.
⎣ ⎦
1. Set up a table
2. Determine the length of each interval

26. Example 2
Graph f (x ) = ⎢ x ⎥ +1.
⎣ ⎦
1. Set up a table
2. Determine the length of each interval
3. Choose an integer for one endpoint and
the next integer for the other

27. Example 2
Graph f (x ) = ⎢ x ⎥ +1.
⎣ ⎦
1. Set up a table
2. Determine the length of each interval
3. Choose an integer for one endpoint and
the next integer for the other
4. Determine which endpoint is included by
testing a value in between

28. Example 2
Graph f (x ) = ⎢ x ⎥ +1.
⎣ ⎦
1. Set up a table
2. Determine the length of each interval
3. Choose an integer for one endpoint and
the next integer for the other
4. Determine which endpoint is included by
testing a value in between
5. Finish your table and plot your graph

29. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?

30. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?
⎢p⎥
⎢ ⎥
⎣ 50 ⎦

31. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?
⎢p⎥ ⎢150 ⎥
⎢ ⎥ ⎢ ⎥
⎣ 50 ⎦ ⎣ 50 ⎦

32. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?
⎢p⎥ ⎢150 ⎥
⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥
⎣ ⎦
⎣ 50 ⎦ ⎣ 50 ⎦

33. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?
⎢p⎥ ⎢150 ⎥
⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3
⎣ ⎦
⎣ 50 ⎦ ⎣ 50 ⎦

34. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?
⎢p⎥ ⎢150 ⎥
⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3
⎣ ⎦
⎣ 50 ⎦ ⎣ 50 ⎦
3 rolls

35. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?
⎢p⎥ ⎢150 ⎥ ⎢ 786 ⎥
⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3
⎣ ⎦ ⎢ ⎥
⎣ 50 ⎦ ⎣ 50 ⎦ ⎣ 50 ⎦
3 rolls

36. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?
⎢p⎥ ⎢150 ⎥ ⎢ 786 ⎥
⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3 ⎢ ⎥ = ⎢15.72 ⎥
⎣ 50 ⎦ ⎣ 50 ⎦
⎣ ⎦
⎣ 50 ⎦ ⎣ ⎦
3 rolls

37. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?
⎢p⎥ ⎢150 ⎥ ⎢ 786 ⎥
⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3 ⎢ ⎥ = ⎢15.72 ⎥ =15
⎣ 50 ⎦ ⎣ 50 ⎦
⎣ ⎦
⎣ 50 ⎦ ⎣ ⎦
3 rolls

38. Example 3
Banks often put pennies in rolls of 50. How many full rows
can be made from p pennies? From 150 pennies? From 786
pennies?
⎢p⎥ ⎢150 ⎥ ⎢ 786 ⎥
⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3 ⎢ ⎥ = ⎢15.72 ⎥ =15
⎣ 50 ⎦ ⎣ 50 ⎦
⎣ ⎦
⎣ 50 ⎦ ⎣ ⎦
3 rolls 15 rolls

39. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦

40. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦

41. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
−3 x − 2

42. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
−3 x − 2 1.5 −1.5 ⎢1− (−3)⎥
⎣ ⎦

43. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
−3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦

44. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
−3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦
1.5 −1.5 ⎢1− (−2)⎥
⎣ ⎦

45. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
−3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦
1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦

46. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
−3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥
⎣ ⎦
1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦

47. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
−3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3
⎣ ⎦
1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦

48. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3
⎣ ⎦
−3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦

49. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3
⎣ ⎦
−3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦
−2 < x ≤ −1

50. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3
⎣ ⎦
−3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦
−2 < x ≤ −1 -1.5

51. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3
⎣ ⎦
−3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦
−2 < x ≤ −1 -1.5
−1< x ≤ 0

52. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3
⎣ ⎦
−3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦
−2 < x ≤ −1 -1.5
−1< x ≤ 0 0

53. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3
⎣ ⎦
−3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦
−2 < x ≤ −1 -1.5
−1< x ≤ 0 0
0 < x ≤ −1

54. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3
⎣ ⎦
−3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦
−2 < x ≤ −1 -1.5
−1< x ≤ 0 0
0 < x ≤ −1 1.5

55. Example 4
Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ .
⎣ ⎦
x f (x ) =1.5 −1.5 ⎢1− x ⎥
⎣ ⎦
1.5 −1.5 ⎢1− (−3)⎥ = −4.5
⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3
⎣ ⎦
−3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3
⎣ ⎦
−2 < x ≤ −1 -1.5
−1< x ≤ 0 0
0 < x ≤ −1 1.5

56. Homework

57. Homework
p. 189 #1-24, skip 12, 16
“There ain’t no free lunches in this country. And don’t go
spending your whole life commiserating that you got raw
deals. You’ve got to say, ‘ I think that if I keep working at
this and want it bad enough I can have it.’” - Lee Iacocca