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INTRODUCTORY MATHEMATICALINTRODUCTORY MATHEMATICALANALYSISANALYSISFor Business, Economics, and te Li!e and Socia" Sciences

2011 Pearson Education, Inc.

Chapter 2Chapter 2Functions and GraphsFunctions and Graphs


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To understand what functions and domains are. To introduce different types of functions.

To introduce addition, subtraction, multiplication,

division, and multiplication by a constant. To introduce inverse functions and properties.

To graph equations and functions.

To study symmetry about thex- and y-axis.

To be familiar with shapes of the graphs of sixbasic functions.

Chapter 2: Functions and Graphs

Chapter ObjectivesChapter Objectives


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Functions

Special Functions

ombinations of Functions

!nverse Functions

"raphs in #ectangular oordinates

SymmetryTranslations and #eflections


Chapter OutlineChapter Outline

$.%&

$.$&

$.'&

$.(&

$.)&

$.*&

$.+&


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function assigns each input number to oneoutput number.

The set of all input numbers is the domain ofthe function.

The set of all output numbers is the range.

Equality of Functions

Two functions f and g are equal f g)/%.0omain of f domain ofg1

2.f(x) g(x).


2. Functions2. Functions


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2. Functions

E!ample " #etermining Equality of Functions

Determine which of the following functions are equal.

=

+=

=+

=+=

+

=

%if'%if$&d.

%if2

%if$&c.

$&b.

&%

&%&$&a.

x

xxxk

x

xxxh

xxg

x

xxxf


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2. Functions

E!ample " #etermining Equality of Functions

Solution/

3hen x %,

4y definition, g(x) h(x) k(x)for allx%.Since g%& ', h%& 2 and k%& ', we concludethat

( ) ( )

( ) ( )

( ) ( )%%

,%%

,%%

kf

hf

gf

kh

hg

kg

=

,

,


7/302011 Pearson Education, Inc.


2. Functions

E!ample $ " Finding #omain and Function %alues

5et . ny real number can be usedforx, so the domain of g is all real numbers.

a.Find g(z).

Solution/

b.Find g(r2).

Solution/

c. Find g(x 6 h).Solution/

2( ) 3 5g x x x

= +

2( ) 3 5g z z z= +

2 2 2 2 4 2( ) 3( ) 5 3 5

g r r r r r= + = +

2

2 2

( ) 3( ) ( ) 5

3 6 3 5

g x h x h x h

x hx h x h

+ = + + += + + +




2. Functions

E!ample & " #emand Function

Suose that the equation ! "##$q describes therelationshi between the rice er unit of a certain

roduct and the number of units q of the roduct that

consumers will buy (that is% demand) er week at the

stated rice. &rite the demand function.

Solution/ q

q =%22




2.2 'pecial Functions2.2 'pecial Functions

E!ample " Constant Function

3e begin with constant function.

5et h(x) $. The domain of h is all real numbers.

function of the form h(x) c, where c constant, is

a constant function.

(10) 2 ( 387) 2 ( 3) 2h h h x= = + =




2.2 'pecial Functions

E!ample $ " (ational Functions

E!ample & " )bsolute*%alue Function

a. is a rational function, since thenumerator and denominator are both polynomials.

b. is a rational function, since .

2 6( )

5

x xf x

x

=

+

( ) 2 3g x x= +2 3

2 31

xx

++ =

)bsolute*value functionis defined as , e.g.x

if 0

if 0

x x

x x x

=





E!ample + " Genetics

'wo black igs are bred and roduce exactly fie

offsring. t can be shown that the robability * that

exactly r of the offsring will be brown and the others

black is a function of r %

+n the right side% * reresents the function rule. +n

the left side% * reresents the deendent ariable.'he domain of * is all integers from 2 to ), inclusie.Find the robability that exactly three guinea igs will

be brown.

( )

51 3

5! 4 4( ) 0,1,2,...,5

! 5 !

r r

P r rr r

= =





E!ample + " Genetic

Solution/3 2

1 3 1 95! 120

454 4 64 16

3!2! 6(2) 512(3)P

= ==




2.$ Combinations of Functions2.$ Combinations of Functions

E!ample " Combining Functions

3e define the operations of function as/

( )( ) ( ) ( )

( )( ) ( ) ( )

( )( ) ( ). ( )

( )( ) for ( ) 0

( )

f g x f x g x

f g x f x g x

fg x f x g x

f f xx g x

g g x

+ = + =

=

=

f f(x) 'x 7 % and g(x) x$6 'x, find a. ( )( )

. ( )( ) c. ( )( )

d. ( )

1

e. ( )( )2

f g x

f g xfg x

fx

f x

+

C G




2.$ Combinations of Functions

E!ample " Combining Functions

Solution/2 2

2 2

2 3 2

2

a. ( )( ) ( ) ( ) (3 1) ( "3 ) 6 1

. ( )( ) ( ) ( ) (3 1) ( "3 ) 1

c. ( )( ) ( ) ( ) (3 1)( 3 ) 3 8 3

( ) 3 1d. ( ) ( ) 3

1 1 1 3 1e. ( )( ) ( ( )) (3 1)

2 2 2

f g x f x g x x x x x x

f g x f x g x x x x x

fg x f x g x x x x x x x

f f x xxg g x x x

xf x f x x

+ = + = + = + = = =

= = + = +

= = +

= = =2

Composition

omposite of f with g is defined by ( )( ) ( ( ))f g x f g x=o

Ch t 2 F ti d G h




2.$ Combinations of Functions

E!ample $ " Composition

Solution/

2If ( ) 4 3, ( ) 2 1, and ( ) ,find

a. ( ( ))

. ( ( ( )))

c. ( (1))

F p p p G p p H p p

F G p

F G H p

G F

= + = + =

2 2

2 2

2

a. ( ( )) (2 1) (2 1) 4(2 1) 3 4 12 2 ( )( )

. ( ( ( ))) ( ( ))( ) (( ) )( ) ( )( ( ))

( )( ) 4 12 2 4 12 2

c. ( (1)) (1 4 1 3) (2) 2 2 1 5

F G p F p p p p p F G p

F G H p F G H p F G H p F G H p

F G p p p p p

G F G G

= + = + + + = + + =

= = = =

= + + = + =

= + = = + =

o

o o o o o

o

Ch t 2 F ti d G h




2., -nverse Functions2., -nverse Functions

E!ample " -nverses of inear Functions

n inverse function is defined as 1 1( ( )) ( ( ))f f x x f f x = =

Show that a linear function is one,to,one. Find the

inerse of f(x) ax 6 b and show that it is also linear.

Solution/ssume that f(u) f(), thus .

3e can prove the relationship,

au b av b+ = +

( )( )( ) ( ( ))

ax b b axg f x g f x x

a a

+ = = = =o

( )( ) ( ( )) ( )x bf g x f g x a b x b b xa= = + = + =o

Ch t 2 F ti d G h




2., -nverse Functions

E!ample $ " -nverses /sed to 'olve Equations

-any equations take the form f(x) 2% where f is afunction. f f is a one,to,one function% then the

equation has x f 7%(2) as its unique solution.

Solution/pplying f 7%to both sides gives .

Since , is a solution.

( )( ) ( )1 1 0f f x f =1(0)f

1( (0)) 0f f =

Chapter 2 F nctions and Graphs




2., -nverse Functions

E!ample & " Finding the -nverse of a Function

'o find the inerse of a one,to,one function f % sole

the equation y ! f(x) for x in terms of y obtaining x !

g(y). 'hen f"(x)!g(x). 'o illustrate% find f"(x) if

f(x)!(x ")2% for x / ".

Solution/

5et y x 7 %&$, for x 8 %. Then x 7 % 9y and hence x

9y 6 %. !t follows that f7%

x& 9x 6 %.





2.& Graphs in (ectangular Coordinates2.& Graphs in (ectangular Coordinates

The rectangular coordinate system provides a

geometric way to graph equations in twovariables.

n !*interceptis a point where the graph

intersects thex-axis.0*interceptis vice versa.





2.& Graphs in (ectangular Coordinates

E!ample " -ntercepts and Graph

Find the x, and y,intercets of the grah of y $x 6 ',and sketch the grah.

Solution/

3hen y 2, we have

3henx 2,

30 2 3 so t#at2

x x= + = 2(0) 3 3y= + =






E!ample $ " -ntercepts and Graph

Determine the intercets of the grah of x ', andsketch the grah.

Solution/

There is no y-intercept, becausex cannot be 2.






E!ample + " Graph of a Case*#efined Function

0rah the case,defined function

Solution/

if 0 $ 3

( ) 1 if 3 5

4 if 5 $ 7

x x

f x x x

x

=




1se the receding definition to show that the grah

of y x$is symmetric about the y,axis.

Solution/

3hen a, b& is any point on the graph, .3hen -a, b& is any point on the graph, .

The graph is symmetric about the y,axis.


2.1 'ymmetry2.1 'ymmetry

E!ample " y*)!is 'ymmetry

graph is symmetric about the y*a!is when -a,b& lies on the graph when a, b& does.

2

b a=2 2( )a a b = =



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2.1 'ymmetry

"raph is symmetric about the !*a!is when x, -y&lies on the graph when x, y& does.

"raph is symmetric about the origin when 7x,7y&lies on the graph when x, y& does.

Summary/



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2.1 'ymmetry

E!ample $ " Graphing ith -ntercepts and 'ymmetry

'est y f (x) %7x(

for symmetry about the x,axis%the y,axis% and the origin. 'hen find the intercetsand sketch the grah.



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2.1 'ymmetry

E!ample $ " Graphing ith -ntercepts and 'ymmetry

Solution/#eplace y with :y, not equivalentto equation.

#eplace x with :x, equivalentto equation.

#eplace x with :x and y with :y, not equivalentto

equation.Thus, it is only symmetric about the y*a!is.

!ntercept at

41 0

1 or 1

x

x x == =



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2.1 'ymmetry

E!ample & " 'ymmetry about the ine y 3x

graph is symmetric about the y3xwhen b, a&

and a, b&.

Show that x$6 y$ % is symmetric about the line

y x.Solution/

!nterchanging the roles of x and y produces

y$

6x$

% equivalent to x$

6 y$

%&.!t is symmetricabout y x.



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2.+ 4ranslations and (eflections2.+ 4ranslations and (eflections

* frequently used functions/



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C apte u ct o s a d G ap s

2.+ 4ranslations and (eflections

4asic types of transformation/



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p p

2.+ 4ranslations and (eflections

E!ample " 5ori6ontal 4ranslation

Sketch the grah of y ! (x ").

Solution/