Piecewise Functions

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

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?

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

“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.

Greatest Integer/Floor Function:

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.

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

Graphs of the floor & ceiling functions

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¢

Long D’s Basic Residential Plan

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¢

Postage Stamp Function

How Much Has It Rained?