Chapter 2 - Functions and Graphs

INTRODUCTORY MATHEMATICALINTRODUCTORY MATHEMATICAL
ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences
©2007 Pearson Education Asia
Chapter 2Chapter 2
Functions and GraphsFunctions and Graphs

INTRODUCTORY MATHEMATICAL
ANALYSIS
0. Review of Algebra
1. Applications and More Algebra
2. Functions and Graphs
3. Lines, Parabolas, and Systems
4. Exponential and Logarithmic Functions
5. Mathematics of Finance
6. Matrix Algebra
7. Linear Programming
8. Introduction to Probability and Statistics

9. Additional Topics in Probability
10. Limits and Continuity
11. Differentiation
12. Additional Differentiation Topics
13. Curve Sketching
14. Integration
15. Methods and Applications of Integration
16. Continuous Random Variables
17. Multivariable Calculus
INTRODUCTORY MATHEMATICAL
ANALYSIS

• 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 the x- and y-axis.
• To be familiar with shapes of the graphs of six
basic functions.
Chapter 2: Functions and Graphs
Chapter ObjectivesChapter Objectives

Functions
Special Functions
Combinations of Functions
Inverse Functions
Graphs in Rectangular Coordinates
Symmetry
Translations and Reflections
Chapter OutlineChapter Outline
2.1)
2.2)
2.3)
2.4)
2.5)
2.6)
2.7)

• A function assigns each input number to one
output number.
• The set of all input numbers is the domain of
the function.
• The set of all output numbers is the range.
Equality of Functions
• Two functions f and g are equal (f = g):
1.Domain of f = domain of g;
2.f(x) = g(x).
2.1 Functions2.1 Functions

2.1 Functions
Example 1 – Determining Equality of Functions
Determine which of the following functions are equal.



=
≠+
=



=
≠+
=
+=
−
−+
=
1if3
1if2
)(d.
1if0
1if2
)(c.
2)(b.
)1(
)1)(2(
)(a.
x
xx
xk
x
xx
xh
xxg
x
xx
xf

2.1 Functions
Example 1 – Determining Equality of Functions
Solution:
When x = 1,
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
( ) ( )
( ) ( )
( ) ( )11
,11
,11
kf
hf
gf
≠
≠
≠
kh
hg
kg
≠
≠
=
,
,

2.1 Functions
Example 3 – Finding Domain and Function Values
Let . Any real number can be used
for x, so the domain of g is all real numbers.
a. Find g(z).
Solution:
b. Find g(r2
).
Solution:
c. Find g(x + h).
Solution:
2
( ) 3 5g x x x= − +
2
( ) 3 5g z z z= − +
2 2 2 2 4 2
( ) 3( ) 5 3 5g 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
+ = + − + +
= + + − − +

2.1 Functions
Example 5 – Demand Function
Suppose that the equation p = 100/q describes the
relationship between the price per unit p of a certain
product and the number of units q of the product that
consumers will buy (that is, demand) per week at the
stated price. Write the demand function.
Solution: p
q
q =
100


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, is
a constant function.
(10) 2 ( 387) 2 ( 3) 2h h h x= − = + =

2.2 Special Functions
Example 3 – Rational Functions
Example 5 – Absolute-Value Function
a. is a rational function, since the
numerator and denominator are both polynomials.
b. is a rational function, since .
2
6
( )
5
x x
f x
x
−
=
+
( ) 2 3g x x= +
2 3
2 3
1
x
x
+
+ =
Absolute-value function is defined as , e.g.x
if 0
if 0
x x
x
x x
≤ 
=  
− < 

Example 7 – Genetics
Two black pigs are bred and produce exactly five
offspring. It can be shown that the probability P that
exactly r of the offspring will be brown and the others
black is a function of r ,
On the right side, P represents the function rule. On
the 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 will
be brown.
( )
5
1 3
5!
4 4
( ) 0,1,2,...,5
! 5 !
r r
P r r
r r
−
   
 ÷  ÷
   = =
−

Example 7 – Genetic
Solution:
3 2
1 3 1 9
5! 120
454 4 64 16
3!2! 6(2) 512
(3)P
      
 ÷  ÷  ÷ ÷
      = ==

2.3 Combinations of Functions2.3 Combinations of Functions
Example 1 – Combining 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 x
x g x
g g x
+ = +
− = −
=
= ≠
If 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
f
x
f x
+
−

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 ) 1
c. ( )( ) ( ) ( ) (3 1)( 3 ) 3 8 3
( ) 3 1
d. ( )
( ) 3
1 1 1 3 1
e. ( )( ) ( ( )) (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 x
x
g g x x x
x
f x f x x
+ = + = − + = + −
− = − = − − = − −
= = − + = + −
−
= =
+
−
= = − =
2
Composition
• Composite of f with g is defined by ( )( ) ( ( ))f g x f g x=o

2.3 Combinations of Functions
Example 3 – Composition
Solution:
2
If ( ) 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 p
G 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

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 the
inverse of f(x) = ax + b and show that it is also linear.
Solution:
Assume that f(u) = f(v), thus .
We can prove the relationship,
au b av b+ = +
( )
( )( ) ( ( ))
ax b b ax
g f x g f x x
a a
+ −
= = = =o
( )( ) ( ( )) ( )
x b
f g x f g x a b x b b x
a
−
= = + = − + =o

2.4 Inverse Functions
Example 3 – Inverses Used to Solve Equations
Many equations take the form f(x) = 0, where f is a
function. 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 −
=

2.4 Inverse Functions
Example 5 – Finding the Inverse of a Function
To find the inverse of a one-to-one function f , solve
the equation y = f(x) for x in terms of y obtaining x =
g(y). Then f−1
(x)=g(x). To illustrate, find f−1
(x) if
f(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 f−1
(x) = √x + 1.

2.5 Graphs in Rectangular Coordinates2.5 Graphs in Rectangular Coordinates
• The rectangular coordinate system provides a
geometric way to graph equations in two
variables.
• An x-intercept is a point where the graph
intersects the x-axis. Y-intercept is vice versa.

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:
When y = 0, we have
When x = 0,
3
0 2 3 so that
2
x x= + = −
2(0) 3 3y = + =

Example 3 – Intercepts and Graph
Determine the intercepts of the graph of x = 3, and
sketch the graph.
Solution:
There is no y-intercept, because x cannot be 0.

Example 7 – Graph of a Case-Defined Function
Graph the case-defined function
Solution:
if 0 < 3
( ) 1 if 3 5
4 if 5 < 7
x x
f x x x
x
≤

= − ≤ ≤
 ≤

Use the preceding definition to show that the graph
of y = x2
is symmetric about the y-axis.
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.
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.
2
b a=
2 2
( )a a b− = =

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:

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.

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 to
equation.
Thus, it is only symmetric about the y-axis.
Intercept at
4
1 0
1 or 1
x
x x
− =
= = −

2.6 Symmetry
Example 5 – Symmetry about the Line y = x
• A graph is symmetric about the y = x when (b, a)
and (a, b).
Show that x2
+ y2
= 1 is symmetric about the line
y = x.
Solution:
Interchanging the roles of x and y produces
y2
+ x2
= 1 (equivalent to x2
+ y2
= 1).
It is symmetric about y = x.

2.7 Translations and Reflections2.7 Translations and Reflections
• 6 frequently used functions:

2.7 Translations and Reflections
• Basic types of transformation:

2.7 Translations and Reflections
Example 1 – Horizontal Translation
Sketch the graph of y = (x − 1)3
.
Solution:

Chapter 2 - Functions and Graphs

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