1. Step functions Greatest integer functions Piecewise functions
2.6 Special Functions
Step functions
Greatest integer functions
Piecewise functions

2. The Constant Function Here f(x) is equal to one number. f(x) = 3.
Have we seen
this before?

3. Absolute Value function: f(x) = | x |
Let plot some points
x f(x)
0 0
1 1
-1 1
2 2

4. Absolute Value function: f(x) = | x |
Let plot some points
x f(x)
0 0
1 1
2 2
Shape V for victory

5. Lets graph f(x) = - | x – 3| x - | x – 3| f(x)
0 - | 0 – 3| = - | - 3| (0, - 3)
1 - | 1 – 3| = - | - 2| - 2 (1, - 2)
2 - | 2 – 3| = - | - 1| - 1 (2, - 1)
3 - | 3 – 3| = - | - 0| (3, 0)
4 - | 4 – 3| = - | 1 | (4, - 1)
5 - | 5 – 3| = - | 2 | (5, - 2)

6. Lets graph f(x) = - | x – 3| (0, - 3) (1, - 2) (2, - 1) (3, 0)
(4, - 1)
(5, - 2)

8. CW 2-5
Page 104
#8-11

9. Piecewise Functions
Graphing different functions over different parts of the graph.
One part tells you what to graph, then where to graph it. What to graph Where to graph

10. Piecewise Functions 2 is where the graph changes.
Less then 2 uses 3x + 2
Greater then 2 uses x - 3

11. We can and should put in a few x into the function
If f(0) we use 3x + 2, then 3(0) + 2 = 2
If f(3) we use x – 3,
then (3) – 3 = 0
The input tell us what function to use.

12. We can and should put in a few x into the function
If we want to find out what f(2) = we use both equations, but leaving an open space on the graph for the point in the function 3x + 2.
Why?

13. We can and should put in a few x into the function
f(2) in 3x + 2;
3(2) + 2 = 8
Graph an open point
at (2,8).
f(2) in x – 3
(2) – 3 = -1
Graphs a filled in point at (2, -1)

14. Piecewise Functions
So put in an x where the domain changes and one point higher
and lower
(2, 8)
(2, -1)

15. Graph the piecewise function

17. Write the Piecewise-defined Function

18. CW 2-6 (Cont.)
Page 105
# 12-16

19. HW 2-6
#’s 17-19, 24-30