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INTRODUCTORY MATHEMATICAL ANALYSISINTRODUCTORY MATHEMATICAL ANALYSISFor Business, Economics, and the Life and Social Sciences

Chapter 2Chapter 2Functions and GraphsFunctions and Graphs

2011 Pearson Education, Inc.


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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

Chapter 2: Functions and Graphs

Chapter ObjectivesChapter Objectives


. To introduce inverse functions and properties.

To graph equations and functions.

To study symmetry about the x- and y-axis.

To be familiar with shapes of the graphs of six

basic functions.


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Functions

Special Functions

Combinations of Functions

Inverse Functions


Chapter OutlineChapter Outline

2.1)

2.2)

2.3)

2.4


Graphs in Rectangular Coordinates

Symmetry

Translations and Reflections

2.8) Functions of Several Variables

2.5)

2.6)

2.7)


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A function assigns each input number to oneoutput number.

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


2.1 Functions2.1 Functions


.Equality of Functions

Two functions f and g are equal (f = g):

1.Domain of f = domain of g;

2.f(x) = g(x).


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Chapter 2: Functions and Graphs2.1 Functions

Example 1 Determining Equality of Functions

Determine which of the following functions are equal.

+=

+

=

2)(b.

)1(

)1)(2(

)(a.

xxg

x

xx

xf


=

+=

=

+=

1if31if2)(d.

1if01if2)(c.

xxxxk

xxxxh


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Example 1 Determining Equality of Functions

Solution:When x = 1,

( ) ( )

( ) ( )( ) ( )11

,11

,11

kf

hf

gf


By definition, g(x) = h(x) = k(x) for all x 1.Since g(1) = 3, h(1) = 0 and k(1) = 3, we conclude

that

kh

hg

kg

=

,

,


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Example 3 Finding Domain and Function Values

Let . Any real number can be usedfor x, so the domain of g is all real numbers.

a. Find g(z).Solution:

2( ) 3 5g x x x= +

2( ) 3 5g z z z= +


b. Find g(r2).Solution:

c. Find g(x + h).Solution:

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

+ = + + +

= + + +


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Example 5 Demand Function

Suppose that the equation p = 100/q describes therelationship between the price per unit p of a certain

product and the number of units q of the product thatconsumers will buy (that is, demand) per week at thestated price. Write the demand function.


Solution: pq

q =100

a


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2.2 Special Functions2.2 Special Functions

Example 1 Constant Function

We begin with constant function.

Let h(x) = 2. The domain of h is all real numbers.


A function of the form h(x) = c, where c = constant, isa constant function.

x= = =


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2.2 Special Functions

Example 3 Rational Functions

a. is a rational function, since the

numerator and denominator are both polynomials.

b. is a rational function, since .

2 6( )

5

x xf x

x

=

+

( ) 2 3g x x= + 2 32 31

xx ++ =


Absolute-value function is defined as , e.g.x

if 0

if 0

x xx

x x

=


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Example 7 Genetics

Two black pigs are bred and produce exactly fiveoffspring. It can be shown that the probability P that

exactly r of the offspring will be brown and the othersblack is a function of r ,

51 3

5!

r r


On the right side, P represents the function rule. Onthe left side, P represents the dependent variable.

The domain of P is all integers from 0 to 5, inclusive.Find the probability that exactly three guinea pigs willbe brown.

( )( ) 0,1,2,...,5! 5 !P r rr r= =


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Example 7 Genetic

Solution:

3 21 3 1 9

5! 120454 4 64 16

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

= ==



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2.3 Combinations of Functions2.3 Combinations of Functions

We 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

+ = +

=

=

=


Example 1 Combining FunctionsIf f(x) = 3x 1 and g(x) = x2 + 3x, find

a. ( )( )

b. ( )( )

c. ( )( )

d. ( )g

1 e. ( )( )2

f g x

f g x

fg x

fx

f x

+


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2.3 Combinations of Functions

Example 1 Combining Functions

Solution:2 2

2 2

2 3 2

2

a. ( )( ) ( ) ( ) (3 1) ( +3 ) 6 1

b. ( )( ) ( ) ( ) (3 1) ( +3 ) 1c. ( )( ) ( ) ( ) (3 1)( 3 ) 3 8 3

( ) 3 1d. ( )

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

f g x f x g x x x x xfg x f x g x x x x x x x

f f x xx

+ = + = + = +

= = =

= = + = +

= =


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

2 2 2

xf x f x x

= = =

2

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


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2.3 Combinations of Functions

Example 3 Composition

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

a. ( ( ))

b. ( ( ( )))c. ( (1))

F p p p G p p H p p

F G p

F G H pG F

= + = + =


2 2

2 2

2

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

b. ( ( ( ))) ( ( ))( ) (( ) )( ) ( )( ( ))

( )( ) 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


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One-to-one function

A function f that satisfies

For all a and b, if f(a)=f(b) then a=b

is called a one-to-one function

Or


For all a and b, if ab then f(a)f(b)

Example

f(x)=x2, then f(-1)=f(1)=1 and -11 show thatthe squaring function is not one-to-one.


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2.4 Inverse Functions2.4 Inverse Functions

Example 1 Inverses of Linear Functions

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

Show that a linear function is one-to-one. Find theinverse of f(x) = ax + b and show that it is also linear.


Assume that f(u) = f(v), thus .

We 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


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2.4 Inverse Functions

Example 3 Inverses Used to Solve Equations

Many equations take the form f(x) = 0, where f is afunction. If f is a one-to-one function, then the

equation has x = f 1(0) as its unique solution.

Solution:


Applying f 1 to both sides gives .

Since , is a solution.

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

=

1(0)f

1( (0)) 0f f =


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2.4 Inverse Functions

Example 5 Finding the Inverse of a Function

To find the inverse of a one-to-one function f , solvethe equation y = f(x) for x in terms of y obtaining x =

g(y). Then f1

(x)=g(x). To illustrate, find f1

(x) iff(x)=(x 1)2, for x 1.


Solution:Let y = (x 1)2, for x 1. Then x 1 = y and hence

x = y + 1. It follows that f1(x) = x + 1.


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2.5 Graphs in Rectangular Coordinates2.5 Graphs in Rectangular Coordinates

The rectangular coordinate system provides ageometric way to graph equations in two

variables. An x-intercept is a point where the graph

intersects the x-axis. Y-interce t is vice versa.



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2.5 Graphs in Rectangular Coordinates

Example 1 Intercepts and Graph

Find the x- and y-intercepts of the graph of y = 2x + 3,and sketch the graph.

Solution:

3


,

When x = 0,

22(0) 3 3y = + =


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Example 3 Intercepts and Graph

Determine the intercepts of the graph of x = 3, andsketch the graph.

Solution:

There is no -interce t because x cannot be 0.


Ch 2 F i d G h


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Example 7 Graph of a Case-Defined Function

Graph the case-defined function

if 0 < 3

( ) 1 if 3 54 if 5 < 7

x x

f x x xx

=


Ch t 2 F ti d G h


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Use the preceding definition to show that the graphof = x2 is s mmetric about the -axis.


2.6 Symmetry2.6 Symmetry

Example 1 y-Axis Symmetry

A graph is symmetric about the y-axis when (-a,b) lies on the graph when (a, b) does.


Solution:

When (a, b) is any point on the graph, .

When (-a, b) is any point on the graph, .

The graph is symmetric about the y-axis.

2b a=

2 2( )a a b = =



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2.6 Symmetry

Graph is symmetric about the x-axis when (x, -y)lies on the graph when (x, y) does.

Graph is symmetric about the origin when (x,y)

lies on the graph when (x, y) does.

Summary:




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2.6 Symmetry

Example 3 Graphing with Intercepts and Symmetry

Test y = f (x) = 1 x4 for symmetry about the x-axis,the y-axis, and the origin. Then find the intercepts

and sketch the graph.




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2.6 Symmetry

Example 3 Graphing with Intercepts and Symmetry

Solution:

Replace y with y, not equivalent to equation.

Replace x with x, equivalent to equation.Replace x with x and y with y, not equivalent toequation.


Thus, it is only symmetric about the y-axis.

Intercept at 41 0

1 or 1

x

x x

=

= =



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2.6 Symmetry

Example 5 Symmetry about the Line y = x

A graph is symmetric about the y= xwhen (b, a)and (a, b).

Show that x2 + y2 = 1 is symmetric about the liney = x.


Solution:Interchanging the roles of x and y produces

y2 + x2 = 1 (equivalent to x2 + y2 = 1).

It is symmetric about y = x.



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2.7 Translations and Reflections2.7 Translations and Reflections

6 frequently used functions:




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p p

2.7 Translations and Reflections

Basic types of transformation:




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p p

2.7 Translations and Reflections

Example 1 Horizontal Translation

Sketch the graph of y = (x 1)3.

Solution:



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2.8 Functions of Several Variables

For any three sets X, Y and Z, the notation ofa function f: XYZ

fis simply a rule which assign is assigns toeach element (x,y) in XY at most one


, , .

Example

f(x,y)=x+y is a function of two variables.

f(1,1)=1+1=2f(2,3)=2+3=5


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Graphing a Plane

In space, the graph of an equation of theform

Ax+By+Cz+D=0

where D is a constant and A, B, and C areconstants that are not all zero, is a plane.


same line) determine a plane, a convenientway to sketch a plane is to first determinethe points, if any, where the plane intercept

the x-, y-, and x-axes. These points arecalled intercepts.


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Example 1Sketch the plane 2x+3y+z=6.

The plane intersects the x-axis when y=0 and

z=0. Thus 2x=6 which gives x=3.

Similarly, if x=z=0, then y=2;


if x=y=0, then z=0.

Therefore, the intercepts are (3,0,0), (0,2,0)

and (0,0,6). After these points are plotted, aplane is passed through them.

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