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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

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Piecewise Defined Functions-

when a function uses different formulas on different parts of its domain.

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Piecewise Defined Functions-

when a function uses different formulas on different parts of its domain.

An example:

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Here is a graph of this piecewise function:

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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

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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

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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

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But you lose the real meaning by doing so.

99 for 0 < t 20( )

99 7( 20) for t > 20C f t

t

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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.

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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.

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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.

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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)

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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?

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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

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The Absolute Value Function

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The Absolute Value Function(defined piecewise):

for 0( )

for < 0

x xf x x

x x

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Some examples:

0 ?

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Some examples:

0 0

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Some examples:

5 ?

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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

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This completes Section 2.3

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