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Piecewise Functions
2007 MS Mathematics Framework
• 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.
y   x 
The ceiling function (or least integer function)
will round any number up to the nearest integer.
 4.7   5
  4 .7    4
“Step functions” are sometimes used to describe real-life
situations.
y   x 
The greatest integer function (or floor function) will
round any number down to the nearest integer.
 4 .7   4
  4 .7    5
Greatest Integer/Floor Function:
y  greatest integer  x
y   x  , y   x  , y  x
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] = 3
[15.25] = 15
[4] = 4
[4.999] = 4
[0.14] = 0
[-1.5] = -2
Graphs of the floor & ceiling functions
y   x 
y   x 
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
f (1 minute, 44 seconds) 
65¢
0<x<1
40¢
f (3 minutes, 2 seconds) 
115¢
1<x<2
65¢
2<x<3
90¢
3<x<4
115¢
4<x<5
140¢
5<x<6
165¢
f (4 minutes, 58 seconds) 
140¢
f (7 minute, 30 seconds) 
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
Examples
f (.78) 
f (2.11) 
f (5.01) 
0<x<1
32¢
1<x<2
55¢
78¢
2<x<3
78¢
147¢
3<x<4
101¢
4<x<5
124¢
5<x<6
147¢
32¢
Postage Stamp Function
How Much Has It Rained?
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