Teacher: Liubiyu

Chapter 1-2

Contents

§1.2 Elementary functions and graph

§2.1 Limits of Sequence of number

§2.2 Limits of functions

§1.1 Sets and the real number

§2.3 The operation of limits

§2.4 The principle for existence of limits

§2.5 Two important limits

§2.6 Continuity of functions

§2.7 Infinitesimal and infinity quantity, the order for infinitesimals

Purpose of teachingPurpose of teaching

(1) To understand the concept of a function, to know the ways of representing a function and how to set the functional relationships based on the practical problem; (2) To know the bounded functions, monotone functions, odd function and even function, periodic functions

(3) To understand the concept of composition of functions and piecewise functions, to know the concept of inverse functions and implicit functions

Purpose of teachingPurpose of teaching

(4) To master properties of basic elementary functions and graph

(5) To know how to construct function represent about simple application problems

New WordsNew Words

increasing 递增 递增 range 值域 值域 decreasing 递减递减

independent variable 自变量 自变量 monotonic 单调的 单调的

dependent variable 因变量 因变量 functions 函数函数

domain of definition 定义域 定义域 odd functions 奇函数奇函数

even functions 偶函数偶函数 sum 和和

difference 差 差 integration 积分学积分学

calculus 微积分 微积分 differentiation 微分学微分学

composite functions 复合函数复合函数 product 积 积

piecewise defined function 分段函数分段函数

inverse functions 反函数 反函数 quotient 商商

elementary functions 初等函数初等函数

implicit functions 隐函数 隐函数 power functions 幂函数幂函数

exponential functions 指数函数指数函数

logarithm functions 对数函数对数函数

trigonometric functions 三角函数三角函数

inverse trigonometric functions 反三角函数反三角函数

§1.2 Elementary functions and graph§1.2 Elementary functions and graph

This section develops the notion of a function, and shows how functions can be built up from simpler functions.

11 、、 The concept of a functionThe concept of a function 11 、、 The concept of a functionThe concept of a function

Definition 1Definition 1

Let and be sets of numbers. If for every , there

is a unique corresponding to according to some

determined rule , then is called a function, denoted

by , , or : , .

X Y x X

y Y x

f f

y f x x X f x y f x x X

is called the argument or independent variable, is

called the value of the function at , or dependent

x y

x

variable. The set is called the domain of definition

of the function . The set of all the value of the function

is called the range of and it is a subset of .

X

f

f Y

X function fx y

Y

Domain Range

NotationsNotations

(1) The domain of definition and rule are the two important factors to determine the function. The former describes the region of existence of the function, and the latter gives the method for determining the corresponding elements of the set Y from the elements of the set X.

A function is completely determined by these two factors and is independent of the forms of the expression and the kind of elements contained in the set.

(2) When the function is given by a formula, the domain is usually understood to consist of all the numbers for which the formula is defined.

Example 1Example 1

2

Find the domain of the function

1 4

1y x

x

SolutionSolution

2

2

1For 4 to be meaningful, the square roots

1

of 4 and 1 must make sense, and

xx

x x

thatsuch numbers all of

consists domain theThus, time.same at the 01

x

x

24 0

1 0

or, equivalently, 1 2

That is, the domain is the interval (1,2].

x

x

x

Example 2Example 2

Which is the domain of definition of the following function

1 ( ) ( )?

11

11

f x

x

1,0but , (D) ;2

1,1,0but , C)(

;01

1but , (B) ;0but , (A)

xRxxRx

xRxxRx

Solution

1 1For to be meaningful, 0,1 0

11

11

1and 1 0, that is, the domain of definition is

11

xx

x

x

2

1,1,0 x

Example 3Example 3

Judge whether the following pair of functions are equal?

xxgxxf lg2)( and lg)( 1 2

2)( and )( 2 xxgxxf

33 34 1)( and )( 3 xxxgxxxf

SolutionSolution

(1) The function and are not the same function

because the domain of definition of is different from

.

f x g x

f x

g x

(2) The function and are not the same function

because the formula of is different from .

f x g x

f x g x

(3) The function and are the same function,

because the formula and the domain of definition of

and are the same.

f x g x

f x

g x

22 、 、 Ways of representing a function Ways of representing a function 22 、 、 Ways of representing a function Ways of representing a function

To express a function is mainly to express its corresponding rule. There are many methods to express the corresponding rule, the following three are often used.

(1) Analytic representation(1) Analytic representation

Many functions are given by an analytic representation. For example, the functions are given in example 1 and 2.

(2) Method of tabulation(2) Method of tabulation

Sometimes, a function is given by a table that lists the independent variable and its corresponding dependent variable. For example

x 20 30 40 50 60 70y 20 41 72 118 182 266

(3) Method shown by graph(3) Method shown by graph

The relation between y and x is shown by a graph. For example, the temperature curve recorded by some instruments expresses the relation between the temperature and time.

t

T

0 5 10 20

0.5

1.0

1.5

2.0

15

33 、 、 Properties of functions Properties of functions 33 、 、 Properties of functions Properties of functions

(1) Bounded Functions

Let be the domain of definition of the function

and the set . If there exists a positive number ,

such that ( ) for . Then the function

is bounded on .

D y f x

X D M

f x M x X

y f x X

(2) Monotone Functions

Let be the domain of the function and the

set ,

D y f x

I D

1 2 1 2 1 21 If ( ) ( ) for all , , and ,f x f x x x I x x

then is called an increasing function on ;f x I

; on function decreasinga called is then

, and ,, allfor )()( If 2 212121

Ixf

xxIxxxfxf

(3) Odd Functions and Even Functions(3) Odd Functions and Even Functions

Let ( , )( 0) be the domain of definition of the

function ,

a a a

y f x

function. odd an called is

then),,( allfor )()( If 1 xfaaxxfxf

2 If ( ) ( ) for all ( , ), then is

called an even function.

f x f x x a a f x

NotationsNotations

(3) The graph of an odd function is symmetric with respect to the origin, and graph of an even function is symmetric with respect to the y-axis.

y

x

)( xf

)( xfy

o x-x

)( xf)( xf

y

x

)( xf

o x- x

)( xfy

(3) Periodic Functions(3) Periodic Functions

Let be the domain of definition of the function

( ), if there exists the number 0 such that

( ) ( ), for all , , then is

called a periodic function and is called a period of

.

D

y f x T

f x T f x x D x T D f x

T

f x

2

T

2

3T2

3T

2

T

Example 4Example 4

even?or odd is

)1ln()( function he whether tDetermine 2xxxf

Solution

also ),,( on defined is function This

)()1ln(

)1

1ln()1ln()(

2

2

2

xfxx

xxxxxf

function. odd an is Therefore, xf

Example 5Example 5

function periodic a is that provethen

,),( allfor ,)()(

such that 0constant a exists thereIf

xf

xxfcxf

c

ProofProof

Since ( ) ( ) for all ( , ), then f x c f x x

)()(])[()2( xfcxfccxfcxf

function. periodica is )( Thus xf

44 、 、 Operation rule for functions Operation rule for functions 44 、 、 Operation rule for functions Operation rule for functions

(1) Rational operation rule for functions(1) Rational operation rule for functions

The sum, difference, product and quotient of two functions

and are defined by the following rules on the

domain of definition , where and are the

domain of definition of the function

f g f g

f x g x

D D D D

and ,

respectively.

f x g x

Definition 2Definition 2

),()())(( xgxfxgf

),()())(( xgxfxgf

),()())(( xgxfxfg

( )( ) ( ( ) 0)

( )

f f xx g x

g g x

Definition 3Definition 3 (composition of functions)

(2) Composition of operations rule for functions(2) Composition of operations rule for functions

Let , and be sets. Let be a function from to

, and let be a function from to , then the function

that assigns each element in to the element

in is called the composition of a

X Y Z g X

Y f Y Z

x X f g x

Z f

nd . It is denoted

( is read as " circle " or as " composed

with "), or, .

g

f g f g f g f

g y f g x f g x

X

Y

Z

xg xg

f xgf

NotationsNotations

(4) The range ( ) of must be the subset of the domain

of , that is ( ) , and ( ) is called the

middle variable.

R g g

Y f R g Y u g x

(5) The above figure depicts the notion of a composite

function.

Example 6Example 6

SolutionSolution2 2Since, (sin ) 1 cos 2cos 2(1 sin )

2 2 2

x x xf x

)2

(cos find ,cos1)2

(sin that Supposex

fxx

f

2so, we have ( ) 2(1 ).f u u

xxxx

f cos12

sin2)2

cos1(2)2

(cos

Therefore,

22

Example 7Example 7

xfx

xx

xf find ,1

)1

( that Suppose2

2

SolutionSolution2 2

2

1 1 1Since, ( ) ( ) 2f x x x

x x x

2)( Therefore, 2 xxf

(3) inverse functions(3) inverse functions

1

Suppose that ( ) is the domain of definition of the

function , is the range of , the inverse function

from to is a rule (that is ) for

assigning one element to each element

D f

f R f f

f R f D f x f y

y D f

x R

.f

Definition 4Definition 4 ( inverse functions)

NotationsNotations

1

1

(6) The domain of definition of is the range of its

inverse function , and the domain of definition of

is the range of .

f

f

f f

1 1 1

(7) For a function , if its inverse function exists, then

, ,

, ,

f

y f x x D f y R f

x f y D f R f x R f D f

1

Hence the function and its inverse function

are represented by the same relation, so

their graph is the same curve in the plane.

y f x

x f y

xOy

1

1

(8) But we are accustomed to express the independent

variable by , and the dependent variable by , so that

the inverse function is usually expressed by

. In this case, the graph of the fu

x y

x f y

y f x

1

nction

and the inverse function are symmetric with

respect to the line .

y f x

y f x

y x

)(xfy

x

y

o

),( abQ

),( baP

xfy 1

Example 8Example 8

Find the inverse function of the following functions

23

32 1

x

xy 1)2ln( 2 xy

SolutionSolution

1 First change the position of and in the analytic

2 3representation, we obtain, , that is,

3 2

x y

yx

y

3 2 2 3xy x y

3 2 2 3Thus, is the inverse function of

2 3 3 2

x xy y

x x

12 Since ln( 2) 1, that is, 2 exx y y 1Thus, e 2 is the inverse function of

ln( 2) 1

xy

y x

55 、 、 Piecewise defined function Piecewise defined function 55 、 、 Piecewise defined function Piecewise defined function

It should be noted that the analytic representation of a function sometimes consists of several components on different subsets of the domain of definition of the function. A function expressed by this kind of representations is called a piecewise defined function.

Example 9Example 9 ( Sign function )

0,1

0,0

0,1

sgn

x

x

x

xy

Its domain of definition is ( ) ( , ), its range

is ( ) { 1,0,1}

D f

R f

Example 10Example 10 ( The greatest integer function )

The function whose value at any number is the largest

integer smaller than or equal to is called the greatest

integer function, denoted by

[ ]( )

x

x

y x x R

3][,1]2[,00,3]5.2[ instance For

( The Dirichlet’s function )

number rationalri an is,0

number rationala is,1)(

x

xxDy

Its domain of definition is ( ) ( , ) and its

range is ( ) {0,1}

D f

R f

Example 11Example 11

The function whose value at any rational number is 1

and at any irrational number is 0 is called the Dirichlet

function, denoted by

Example 12Example 12 ( Integer variable function )

The function whose domain of definition is the set of

positive integers is called an integer variable function,

and denoted by ,y f n n N

1 2

If we rewrite , , then the integer variable

function can be also denoted as follows:

, , , ,

For this reason, an integer variable function is also called

a seque

n

n

f n a n N

a a a

nce.

SolutionSolution

0,1

0,0

0,1

1e,1

1e,0

1e,1

))((

x

x

x

xgfx

x

x

1,e

1,1

1,e

))((1 x

x

x

xfg

,e)( and

1,1

1,0

1,1

)( that Suppose xxg

x

x

x

xf

))(()),(( find xfgxgf

Example 13Example 13

66 、 、 Elementary functions Elementary functions 66 、 、 Elementary functions Elementary functions

Those functions which we have learned in high school,

such that power function: ( is a n umber),

exponential function: ( 0, 1), logarithm

function: log ( 0, 1), trigonometric function:

s

x

a

y x

y a a a

y x a a

y

in , cos , tan , cot , sec ,

csc , inverse trigonometric function: arcsin ,

arccos , arctan , arccot ,are all described

by analytic representation. The above five types of

functions and

x y x y x y x y x

y x y x

y x y x y x

constants are called by the joint name

basic elementary functions.

Definition 5Definition 5 ( elementary function )

A function formed from the six kinds of basic elementary functions by a finite number of rational operations and compositions of functions which can be expressed by a single analytic expression is called an elementary functions.

NotationsNotations

21 2 sin9 For example, , ln 1 are both

arccos

, 0elementary functions. But is not an

sin , 0

x

x

xx x

x

e xf x

x x

elementary function because it can not be described by

one analytic expression.

2ln cos

10 How to decompose a composite function into the

combination of some basic elementary functions is very

important. For example, sin 2 is composed

by the following basic elementary functions:

xy

2sin , , 2 , ln , cos , .wy u u v v w s s t t x

Definition 6Definition 6 ( Hyperbolic function )

Hyperbolic sine: sinh , , Hyperbolic2

x xe ex x R

cosine: cosh , , hyperbolic tangent:2

tanh , , Hyperbolic cotangent:

coth , are called by a joint name

hyperbolic functions.

x x

x x

x x

x x

x x

e ex x R

e ex x Re e

e ex x Re e

There are some identities for hyperbolic functions which are similar to those for trigonometric functions.

sinh sinh cosh cosh sinhx y x y x y cosh cosh cosh sinh sinhx y x y x y

2 2

2 2

cosh sinh 1,sinh 2 2sinh cosh

cosh 2 cosh sinh

x x x x x

x x x

NotationsNotations

11 The inverse function of a hyperbolic function is an

inverse hyperbolic function. They are

1 2

1 2

inverse hyperbolic sine: sinh ln 1 , ,

inverse hyperbolic cosine: cos h ln 1 ,

x x x x R

x x x

1[1, ), inverse hyperbolic tangent: tan h

1 1ln , 1,1

2 1

x x

xx

x

77 、 、 Implicit functions Implicit functions 77 、 、 Implicit functions Implicit functions

In applied problems we often need to investigate a class

of functions in which the correspondence rule between

the dependent variable and the independent variable

is defined by an equation of the fo

y

x

rm , 0,

when , denotes an expression in and .

F x y

F x y x y

Definition 7Definition 7

If there is a function , which is defined on

some interval , such that , 0, , then

, is called a implicit function defined by

the equation , 0.

y f x

I F x f x x I

y f x x I

F x y