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INTRODUCTORY MATHEMATICAL ANALYSISINTRODUCTORY MATHEMATICAL ANALYSISFor Business, Economics, and the Life and Social Sciences
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
Chapter 2: Functions and Graphs
Chapter ObjectivesChapter Objectives
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. 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 2: Functions and Graphs
Chapter OutlineChapter Outline
2.1)
2.2)
2.3)
2.4
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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.
Chapter 2: Functions and Graphs
2.1 Functions2.1 Functions
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.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
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=
+=
=
+=
1if31if2)(d.
1if01if2)(c.
xxxxk
xxxxh
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Chapter 2: Functions and Graphs2.1 Functions
Example 1 Determining Equality of Functions
Solution:When x = 1,
( ) ( )
( ) ( )( ) ( )11
,11
,11
kf
hf
gf
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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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Chapter 2: Functions and Graphs2.1 Functions
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= +
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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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Chapter 2: Functions and Graphs2.1 Functions
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.
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Solution: pq
q =100
a
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Chapter 2: Functions and Graphs
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.
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A function of the form h(x) = c, where c = constant, isa constant function.
x= = =
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Chapter 2: Functions and Graphs
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 ++ =
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Absolute-value function is defined as , e.g.x
if 0
if 0
x xx
x x
=
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Chapter 2: Functions and Graphs
2.2 Special Functions
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
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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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Chapter 2: Functions and Graphs
2.2 Special Functions
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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Chapter 2: Functions and Graphs
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
+ = +
=
=
=
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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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Chapter 2: Functions and Graphs
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
+ = + = + = +
= = =
= = + = +
= =
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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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Chapter 2: Functions and Graphs
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
= + = + =
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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
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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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Chapter 2: Functions and Graphs
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.
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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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Chapter 2: Functions and Graphs
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:
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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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Chapter 2: Functions and Graphs
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.
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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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Chapter 2: Functions and Graphs
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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Chapter 2: Functions and Graphs
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
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,
When x = 0,
22(0) 3 3y = + =
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Chapter 2: Functions and Graphs
2.5 Graphs in Rectangular Coordinates
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.
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Ch 2 F i d G h
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Chapter 2: Functions and Graphs
2.5 Graphs in Rectangular Coordinates
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
=
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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.
Chapter 2: Functions and Graphs
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.
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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 = =
Chapter 2: Functions and Graphs
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Chapter 2: Functions and Graphs
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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Chapter 2: Functions and Graphs
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Chapter 2: Functions and Graphs
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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Chapter 2: Functions and Graphs
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Chapter 2: Functions and Graphs
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.
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Thus, it is only symmetric about the y-axis.
Intercept at 41 0
1 or 1
x
x x
=
= =
Chapter 2: Functions and Graphs
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Chapter 2: Functions and Graphs
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.
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Solution:Interchanging the roles of x and y produces
y2 + x2 = 1 (equivalent to x2 + y2 = 1).
It is symmetric about y = x.
Chapter 2: Functions and Graphs
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Chapter 2: Functions and Graphs
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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Chapter 2: Functions and Graphs
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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
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, , .
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.
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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;
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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.