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
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
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
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