1. Section 2.3 Piecewise Defined Functions 2 Piecewise Defined Functions- when a function uses...
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Transcript of 1. Section 2.3 Piecewise Defined Functions 2 Piecewise Defined Functions- when a function uses...
1
Powerpoint slides copied from or based upon:
Connally,
Hughes-Hallett,
Gleason, Et Al.
Copyright 2007 John Wiley & Sons, Inc.
Functions Modeling Change
A Preparation for Calculus
Third Edition
Section 2.3
Piecewise Defined Functions
2
Piecewise Defined Functions-
when a function uses different formulas on different parts of its domain.
3Page 73
Piecewise Defined Functions-
when a function uses different formulas on different parts of its domain.
An example:
4Page 73
Here is a graph of this piecewise function:
5Page 73
Piecewise Defined Functions-
Another example:
g(x) = x + 1 for x ≤ 2 and
g(x) = 1 for x > 2
Let's graph this function:
6Page 73 Example 1
g(x) = x + 1 for x ≤ 2
andg(x) = 1 for x > 2
7Page 73
A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents for each additional minute or part of a minute. a. Use bracket notation to write a formula for the cost, C, of a call as a function of its length t in minutes.
b. Graph the function.
c. State the domain and range of the function.
8Page 2 Example #2
A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents for each additional minute or part of a minute. a. Use bracket notation to write a formula for the cost, C, of a call as a function of its length t in minutes.
99 for 0 < t 20( )
99 7( 20) for t > 20C f t
t
9Page 74
You can simplify this expression to get:
99 for 0 < t 20( )
99 7( 20) for t > 20
99 for 0 < t 20( )
7 41 for t > 20
C f tt
C f tt
10Page 74
But you lose the real meaning by doing so.
99 for 0 < t 20( )
99 7( 20) for t > 20C f t
t
11Page 74
A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents for each additional minute or part of a minute. b. Graph the function.
12Page 74
A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents for each additional minute or part of a minute. b. Graph the function.
13Page 74
A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents for each additional minute or part of a minute. c. State the domain and range of the function.
14Page 74
A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents for each additional minute or part of a minute. c. State the domain and range of the function.
Domain: t > 0 (call lengths ≤ 0 are silly)
15Page 74
A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents for each additional minute or part of a minute. c. State the domain and range of the function.
Domain: t > 0 (call lengths ≤ 0 are silly)
Range?
16Page 74
A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents for each additional minute or part of a minute.
Range?We can see that the range is:
C ≥ 99
17Page 74
The Absolute Value Function
18Page 76
The Absolute Value Function(defined piecewise):
for 0( )
for < 0
x xf x x
x x
19Page 76
Some examples:
0 ?
20Page 76
Some examples:
0 0
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Some examples:
5 ?
22Page 76
Some examples:
5 ( 5) 5
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Let's form a table and then graph the absolute value function:
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x |x|−3 3−2 2−1 1 0 0 1 1 2 2 3 3
25Page 76
26Page 76
This completes Section 2.3
27Page