1. Piecewise Functions

2. 2007 OK Mathematics Standard
Algebra: e. Write, graph, and analyze linear and nonlinear functions (such as quadratic, greatest integer)

3. Why Piecewise Functions?
What factors are important when determining the price of a phone call?
How do you distinguish the price of a five minute call versus a five and one-half minute call?
How do phone companies use functions to determine your local phone bill?

4. “Step functions” are a type of piecewise functions.
The ceiling function (or least integer function) will round any number up to the nearest integer.

5. “Step functions” are sometimes used to describe real-life situations.
The greatest integer function (or floor function) will round any number down to the nearest integer.

6. Greatest Integer/Floor Function:

7. The TI-84 command for the floor function is int (x).
Graphing the greatest integer function:
The calculator “connects the dots” like a staircase instead of just the steps.
The open and closed circles do not show, but you can just see the steps.

8. Greatest Integer Function
[3.7] =
[15.25] =
[4] =
[4.999] =
[0.14] =
[-1.5] = 3 15 4 4 -2

9. Graphs of the floor & ceiling functions

10. How Much Is That Phone Call?
Phone companies will determine the price of a call by rounding the length of the call to a certain time period (either one minute or six seconds). For instance, a local weekday call on Long D’s Basic Residential plan will cost $0.25 each minute. Suppose that Long D’s also charges a $0.15 connection fee for each call.
f(x) is the cost of placing a phone call that lasts x minutes
Examples
minutes
cost
0<x<1
40¢
1<x<2
65¢
2<x<3
90¢
3<x<4
115¢
4<x<5
140¢
5<x<6
165¢
65¢
115¢
140¢
215¢

11. Long D’s Basic Residential Plan

12. Postage Stamp Function
f(x) is the cost of mailing a letter that weighs x ounces
Weight
cost
0<x<1
32¢
1<x<2
55¢
2<x<3
78¢
3<x<4
101¢
4<x<5
124¢
5<x<6
147¢
Examples
32¢
78¢
147¢

13. Postage Stamp Function

14. How Much Has It Rained?