1. Warm-up: Why is the following not a piecewise-defined function?
Answer: There is an overlap of intervals, and f(4) is not unique. It can be 11 or 6.

2. HW Key: p. 133: all B A D C

3. Greatest Integer Functions

4. Objectives & HW:
The students will be able to graph and analyze Greatest Integer functions.
HW: p. 134: even, 51, 55

5. Step functions Your grades are based on a step function. Grade Scale
Letter grades have the following percentage equivalents: A
B 80-89
C 70-79
D 60-69
E 0- 59

6. Greatest Integer Function is a step function
The function is written as
It is not an absolute value. The function rounds down to the last integer less than or equal to x.

7. It may be helpful to visualize this function a little more clearly by using a number line.1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7
6.31
-6.31
Example: [6.31] = 6
Example: [-6.31] = -7
When you use this function, the answer is the integer on the immediate left on the number line. There is one exception. When the function acts on a number that is itself an integer. The answer is itself.
Example: [5] = 5
Example: [-5] = -5
Example:
Example:

8. Find the value of a number in the Greatest Integer function f(x) =[x]
f(2.7) = 2 f(0.8) = 0 f(- 3.4) = - 4
It rounds down to the last integer less than or equal to x.
Find the value
f( 5.8) = f(⅛) = f(- ⅜) =

9. f(x) = [x] x
f(0) = [0] = 0
f(0.5) = [0.5] = 0
0.5
f(0.7) = [0.7] = 0
0.7
f(0.8) = [0.8] = 0
0.8
f(0.9) = [0.9] = 0
0.9
f(1) = [1] = 1 1
f(1.5) = [1.5] = 1
1.5
f(1.6) = [1.6] = 1
1.6
f(1.7) = [1.7] = 1
1.7
f(1.8) = [1.8] = 1
1.8
f(1.9) = [1.9] = 1
1.9
f(2) = [2] = 2 2
f(-0.5) =[-0.5]=-1
-0.5
f(-0.9) =[-0.9]=-1
-0.9
f(-1) = [-1] = -1 -1

10. A step function graph

11. How to graph a step function; f(x)= [x]
Find the values of x = .., -2, -1, 0, 1, 2, ……
f(-2) = -2
f(-1) = - 1
f(0) = 0
f(1) = 1
f(2) = 2

12. Now lets look at 0.5,1.5, -0.5, -1.5
f(-1.5) = -2 It is the same as f( - 2) = -2
f(-0.5) = f( - 1) = -1
f(0.5) = f(0) = 0
f(1.5) = f(1) = 1
So between 0 and almost 1 it equal 0 f( ) = 0

13. How to show all those number equal 0
A close circle at (0, 0)
and an open circle at (1, 0).
(1, 0)
What happens when x = 1?

14. How to show all those number equal 0
A close circle at (0, 0)
and an open circle at (1, 0).
(1,1) (2,1)
(1, 0)
What happens when x = 1? It jumps to (1,1)

15. Is the step only one unit long?
It will be in f(x) = [ x ].
Here is how I graph them.
Find the fill in circles.
Draw line segments ending in a open circle.

16. How to graph a step function; f(x)= [x]
Find the values of x = .., -2, -1, 0, 1, 2, ……
f(-2) = -2
f(-1) = - 1
f(0) = 0
f(1) = 1
f(2) = 2

17. Graph: f(x) = 2[x]

18. Graph: f(x) = -[x]

19. Graph: f(x) = [-x]

20. Graph: f(x) = [⅓ x]