INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007 Pearson Education Asia Chapter 2 Functions and Graphs

INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences

2007 Pearson Education Asia

Chapter 2 Chapter 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


Chapter 2: Functions and Graphs2.1 Functions

Example 1 – Determining Equality of Functions

Determine which of the following functions are equal.

1 if 3

1 if 2)( d.

1 if 0

1 if 2)( c.

2)( b.

)1(

)1)(2()( a.

x

xxxk

x

xxxh

xxg

x

xxxf



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

,

,



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 5

g r r r r r

2

2 2

( ) 3( ) ( ) 5 3 6 3 5

g x h x h x hx hx h x h



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: pq

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 xf x

x

( ) 2 3g x x 2 3

2 31

xx

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

if 0

if 0

x xx

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.

51 3

5!4 4

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

r r

P r rr r




Example 7 – Genetic

Solution:3 2

1 3 1 95! 120

454 4 64 163!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 xf g x f x g xfg x f x g xf f xx g x

g g x

If f(x) = 3x − 1 and g(x) = x2 + 3x, find a. ( )( ) b. ( )( ) c. ( )( )

d. ( )g1

e. ( )( )2

f g xf g xfg xfx

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 1d. ( )

( ) 31 1 1 3 1

e. ( )( ) ( ( )) (3 1)2 2 2

f g x f x g x x x x x xf g x f x g x x x x xfg x f x g x x x x x x xf f x xx

g g x x xx

f x f x x

2

Composition

• Composite of f with g is defined by ( )( ) ( ( ))f g x f g x



2.3 Combinations of Functions

Example 3 – Composition

Solution:

2If ( ) 4 3, ( ) 2 1, and ( ) ,finda. ( ( ))b. ( ( ( )))c. ( (1))

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



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 axg f x g f x x

a a

( )( ) ( ( )) ( )x b

f g x f g x a b x b b xa



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,

30 2 3 so that

2x 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.

2b a2 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 41 01 or 1x

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:

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INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007 Pearson Education Asia Chapter 2 Functions and Graphs