Slide R.2 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Functions and Models

OBJECTIVES Determine whether or not a

correspondence is a function. Find function values. Graph functions and determine whether or

not a graph is that of a function. Graph functions that are piecewise defined.

R.2


DEFINITION:

A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.

R.2 Functions and Models


a) Microsoft StockDomain RangeMarch 14, 2006 $27.23March 15, 2006 $27.36March 16, 2006 $27.27March 17, 2006 $27.50


Example 1: Determine whether or not each correspondence is a function.

This relationship is a function because each member of the domain corresponds to only one member of the range.


Example 1 (continued): b) Squaring

Domain Range 3 9 4 16 5 –5 25

This relationship is a function because each member of the domain corresponds to only one member of the range, even though two members of the domain correspond to 25.



Example 1 (continued): c) Baseball Teams

Domain RangeArizona DiamondbacksChicago Cubs

White SoxBaltimore Orioles

This relationship is not function because one member of the domain, Chicago, corresponds to two members of the range, Cubs and White Sox.



Example 1 (continued) : d) Baseball Teams

Domain RangeDiamondbacks ArizonaCubs ChicagoWhite SoxOrioles Baltimore

This relationship is a function because each member of the domain corresponds to only one member of the range, even though two members of the domain correspond to Chicago.



Example 3: The squaring function, f , is given by

Find


.)( 2xxf ).( and ),1( ),( ),( ),1( ),3( hxftfkfkfff

f ( 3) ( 3)2 9

f (1) 12 1

f (k) k2

f k k 2 k

f (1 t) (1 t)2 1 2t t 2

f (x h) (x h)2 x2 2xh h2


Example 5: A function subtracts the square of an input from the input. A description of f is given by

Find


.)( 2xxxf

.)()(

and ),( ),4(h

xfhxfhxff

12

44)4( 2

f


Example 5 (concluded): R.2 Functions and Models

22

22

2

2

)2(

)()(

hxhxhx

hxhxhx

hxhxhxf

f (x h) f (x)

h

x h x2 2xh h2 (x x2 )

h

x h x2 2xh h2 x x2

h

h(1 2x h)

h 1 2x h, for h 0


Definition

The graph of a function f is a drawing that represents all the input-output pairs, (x, f (x)). In cases where the function is given by an equation, the graph of a function is the graph of the equation y = f (x).



Example 6: Graph f (x) = x2 –1.



The Vertical Line Test

A graph represents a function if it is impossible to draw a vertical line that intersects the graph more than once.



Example 7: Determine whether each of the following is the graph of a function.



Example 7 (concluded): a) The graph is that of a function. It impossible to draw

a vertical line that intersects the graph more than once.

b) The graph is not that of a function. A vertical line (in fact many) can intersect the graph more than once.

c) The graph is not that of a function.

d) The graph is that of a function.



Example 8: In 2005, Sprint® offered a cellphone calling plan in which a customer’s monthly bill can be modeled by the graph below. The amount of the bill is a function f of the number of minutes of phone use.



Example 8 (continued): Each open dot on the graph indicates that the point at that location is not included in the graph.

a) Under this plan, if a customer uses the phone for 360 min, what is his or her monthly bill?what is his or her monthly bill?

b) If a monthly bill is $55, for how many minutes did b) If a monthly bill is $55, for how many minutes did the customer use the phone?the customer use the phone?



Example 8 (concluded): a) To find the bill for 360 min of use, we locate 360 on the

horizontal axis and move directly up to the graph. We then move across to the vertical axis. Thus, the bill is $40.

b) To find the number of minutes of use when a monthly bill is $55, we locate 55 on the vertical axis, move horizontally to the graph, and note that many inputs correspond to 55. If t represents the number of minutes of use, we must have 600 < t ≤ 700.



Example 10: Graph the function defined as follows:

The function is defined such that g(1) = 3 and for all other x-values (that is, for x ≠ 1), we have g(x) = –x + 2. Thus, to graph this function, we graph the line given by g(x) = –x + 2, but with an open dot at the point above x = 1. To complete the graph, we plot the point (1, 3) since g(1) = 3.


g(x) 3, for x 1

x 2, for x 1


Example 10 (concluded):


Slide R.2 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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