Download 1.2 Elementary functions and graph

Teacher: Liubiyu
Chapter 1-2
Contents
§1.1 Sets and the real number
§1.2 Elementary functions and graph
§2.1 Limits of Sequence of number
§2.2 Limits of functions
§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 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 teaching
(4) To master properties of basic elementary functions
and graph
(5) To know how to construct function represent
about simple application problems
New Words
increasing 递增
range 值域
decreasing 递减
independent variable 自变量
monotonic 单调的
dependent variable 因变量
functions 函数
domain of definition 定义域
even functions 偶函数
odd functions 奇函数
sum 和
difference 差
integration 积分学
calculus 微积分
differentiation 微分学
composite functions 复合函数
product 积
piecewise defined function 分段函数
inverse functions 反函数
quotient 商
elementary functions 初等函数
implicit functions 隐函数
power functions 幂函数
exponential functions 指数函数
logarithm functions 对数函数
trigonometric functions 三角函数
inverse trigonometric functions
反三角函数
§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.
1、The concept of a function
Definition 1
Let X and Y be sets of numbers. If for every x  X , there
is a unique y  Y corresponding to x according to some
determined rule f , then f is called a function, denoted
by y  f  x  , x  X , or f : x  y  f  x  , x  X .
x is called the argument or independent variable, y is
called the value of the function at x, or dependent
variable. The set X is called the domain of definition
of the function f . The set of all the value of the function
is called the range of f and it is a subset of Y .
X
x
Domain
function f
y
Y
Range
Notations
(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 1
Find the domain of the function
1
y  4 x 
x 1
2
Solution
1
For 4  x 
to be meaningful, the square roots
x 1
of 4  x 2 and x  1 must make sense, and
2
x  1  0 at the same time. Thus, the domain consists
of all numbers x such that
4  x 2  0

 x 1  0
or, equivalently, 1  x  2
That is, the domain is the interval (1,2].
Example 2
Which is the domain of definition of the following function
1
f ( x) 
1
1
(
)?
1
1
x
(A) x  R, but x  0;
1
(B) x  R, but 1   0;
x
1
(C) x  R, but x  0,1, ; (D) x  R, but x  0,1
2
Solution
1
For
1
1
1
to be meaningful, x  0,1   0
x
1
1
x
1
and 1 
 0, that is, the domain of definition is
1
1
x
1
x  0,1,
2
Example 3
Judge whether the following pair of functions are
equal?
1 f ( x)  lg x 2 and g ( x)  2 lg x
2
2 f ( x)  x and g ( x)  x
3 f ( x) 
3
x  x and g ( x)  x 3 x  1
4
3
Solution
(1) The function f  x  and g  x  are not the same function
because the domain of definition of f  x  is different from
g  x.
(2) The function f  x  and g  x  are not the same function
because the formula of f  x  is different from g  x  .
(3) The function f  x  and g  x  are the same function,
because the formula and the domain of definition of f  x 
and g  x  are the same.
2、 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
Many functions are given by an analytic representation.
For example, the functions are given in example 1 and 2.
(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
y
20
20
30
41
40
72
50
118
60
182
70
266
(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
2.0
1.5
1.0
0.5
0
5
10
15
20
t
3、 Properties of functions
(1) Bounded Functions
Let D be the domain of definition of the function y  f  x 
and the set X  D. If there exists a positive number M ,
such that f ( x)  M for x  X . Then the function
y  f  x  is bounded on X .
(2) Monotone Functions
Let D be the domain of the function y  f  x  and the
set I  D,
1 If f ( x1 )  f ( x2 ) for all x1, x2  I , and x1  x2 ,
then f  x  is called an increasing function on I ;
2 If
f ( x1 )  f ( x 2 ) for all x1 , x 2  I , and x1  x 2 ,
then f  x  is called a decreasing function on I ;
(3) Odd Functions and Even Functions
Let (a, a)(a  0) be the domain of definition of the
function y  f  x  ,
1 If
f (  x)   f ( x) for all x  ( a, a ), then f x 
is called an odd function.
 2
If f ( x)  f ( x) for all x  ( a, a), then f  x  is
called an even function.
Notations
(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
y  f ( x)
y
y  f ( x)
f ( x)
-x
f ( x )
-x
f ( x)
o
f ( x )
o
x
x
x
x
(3) Periodic Functions
Let D be the domain of definition of the function
y  f ( x), if there exists the number T  0 such that
f ( x  T )  f ( x), for all x  D, x  T  D, then f  x  is
called a periodic function and T is called a period of
f  x.


3T
2
T
2
T
2
3T
2
Example 4
Determine whether t he function f ( x )  ln( x  1  x 2 )
is odd or even?
Solution
This function is defined on ( ,), also
f ( x)  ln(  x  1  x )  ln(
1
2
x  1  x2
  ln( x  1  x 2 )   f ( x)
Therefore, f x  is an odd function.
)
Example 5
If there exists a constant c  0 such that
f ( x  c)   f ( x), for all x  (,),
then prove that f x  is a periodic function
Proof
Since f ( x  c)   f ( x) for all x  (, ), then
f ( x  2c )  f [( x  c )  c ]   f ( x  c )  f ( x )
Thus f ( x ) is a periodic function.
4、 Operation rule for functions
(1) Rational operation rule for functions
Definition 2
The sum, difference, product and quotient of two functions
f  x  and g  x  are defined by the following rules on the
domain of definition D f  Dg , where D f and Dg are the
domain of definition of the function f  x  and g  x  ,
respectively.
( 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)  0)
g
g ( x)
(2) Composition of operations rule for functions
Definition 3 (composition of functions)
Let X , Y and Z be sets. Let g be a function from X to
Y , and let f be a function from Y to Z , then the function
that assigns each element x in X to the element f  g  x  
in Z is called the composition of f and g . It is denoted
f g (f g is read as "f circle g " or as "f composed
with g "), or, y   f g  x   f  g  x   .
g
x
X
g x 
Y
f
f g x 
Z
Notations
(4) The range R( g ) of g must be the subset of the domain
Y of f , that is R( g )  Y , and u  g ( x) is called the
middle variable.
(5) The above figure depicts the notion of a composite
function.
Example 6
x
x
Suppose that f (sin )  1  cos x, find f (cos )
2
2
Solution
x
2 x
2 x
Since, f (sin )  1  cos x  2 cos  2(1  sin )
2
2
2
so, we have f (u )  2(1  u 2 ).
Therefore,
x
2 x
2 x
f (cos )  2(1  cos )  2 sin
 1  cos x
2
2
2
Example 7
1
1
2
Suppose that f ( x  )  x  2 , find f  x 
x
x
Solution
1
1
1 2
2
Since, f ( x  )  x  2  ( x  )  2
x
x
x
Therefore,
f ( x)  x 2  2
(3) inverse functions
Definition 4 ( inverse functions)
Suppose that D( f ) is the domain of definition of the
function f , R  f

f 1 from R  f
to D  f

is the range of f , the inverse function

is a rule (that is x  f  y ) for
assigning one element y  D  f
x  R  f .

to each element
Notations
(6) The domain of definition of f is the range of its
inverse function f 1 , and the domain of definition of
f 1 is the range of f .
(7) For a function f , if its inverse function exists, then
y  f  x, x  D  f , y  R  f 
x  f 1  y  , D  f 1   R  f  , x  R  f 1   D  f 
Hence the function y  f  x  and its inverse function
x f
1
 y  are represented by the same relation, so
their graph is the same curve in the xOy plane.
(8) But we are accustomed to express the independent
variable by x, and the dependent variable by y, so that
the inverse function x  f 1  y  is usually expressed by
y  f 1  x  . In this case, the graph of the function y  f  x 
and the inverse function y  f 1  x  are symmetric with
respect to the line y  x.
y
y  f 1 x 
Q (b, a )
o
y  f ( x)
P (a , b)
x
Example 8
Find the inverse function of the following functions
2x  3
1 y 
3x  2
2 y  ln( x  2)  1
Solution
1 First change the position of x and y in the analytic
2y 3
representation, we obtain, x 
, that is,
3y  2
3xy  2 x  2 y  3
3  2x
2x  3
Thus, y 
is the inverse function of y 
2  3x
3x  2
 2 Since x  ln( y  2)  1, that is, y  2  ex1
Thus, y  e x 1  2 is the inverse function of
y  ln( x  2)  1
5、 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 9 ( Sign function )
 1, x  0

y  sgn x   0, x  0
 1, x  0

Its domain of definition is D( f )  (, ), its range
is R( f )  {1, 0,1}
Example 10 ( The greatest integer function )
The function whose value at any number x is the largest
integer smaller than or equal to x is called the greatest
integer function, denoted by
y  [ x]( x  R)
For instance [ 2.5]  3, 0  0, [ 2 ]  1, [ ]  3
Example 11 ( The Dirichlet’s function )
The function whose value at any rational number is 1
and at any irrational number is 0 is called the Dirichlet
function, denoted by
1, x is a rational number
y  D( x )  
0, x is an ir rational number
Its domain of definition is D( f )  (, ) and its
range is R( f )  {0,1}
Example 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 
If we rewrite f  n   an , n  N  , then the integer variable
function can be also denoted as follows:
a1 , a2 ,
, an ,
For this reason, an integer variable function is also called
a sequence.
 1, x  1

Suppose that f ( x )   0, x  1 and g ( x )  e x ,
 1, x  1

find f ( g ( x )), g ( f ( x ))
Example 13
Solution
 1, e x  1  1, x  0


x
f ( g ( x ))   0, e  1   0, x  0
 1, e x  1  1, x  0


 e, x  1

g ( f ( x ))   1, x  1
e 1 , x  1

6、 Elementary functions
Those functions which we have learned in high school,
such that power function: y  x  (  is a number),
exponential function: y  a ( a  0, a  1), logarithm
x
function: y  log a x( a  0, a  1), trigonometric function:
y  sin x, y  cos x, y  tan x, y  cot x, y  sec x,
y  csc x, inverse trigonometric function: y  arcsin x,
y  arccos x, y  arctan x, y  arc cot x, are all described
by analytic representation. The above five types of
functions and constants are called by the joint name
basic elementary functions.
Definition 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.
Notations


1  2 x sin x
2
, ln x  1  x are both
 9  For example,
arccos x
 ex , x  0
elementary functions. But f  x   
is not an
sin x, x  0
elementary function because it can not be described by
one analytic expression.
10  How to decompose a composite function into the
combination of some basic elementary functions is very
important. For example, y  sin 2
ln cos x 2
is composed
by the following basic elementary functions:
y  sin u, u  v , v  2w , w  ln s, s  cos t , t  x 2 .
Definition 6 ( Hyperbolic function )
x
e e
Hyperbolic sine: sinh x 
, x  R, Hyperbolic
2
x
x
e e
cosine: cosh x 
, x  R, hyperbolic tangent:
2
e x  e x
tanh x  x  x , x  R, Hyperbolic cotangent:
e e
e x  e x
coth x  x  x , x  R are called by a joint name
e e
hyperbolic functions.
There are some identities for hyperbolic functions which
are similar to those for trigonometric functions.
x
sinh  x  y   sinh x cosh y  cosh x sinh y
cosh  x  y   cosh x cosh y  sinh x sinh y
cosh x  sinh x  1,sinh 2 x  2sinh x cosh x
2
2
cosh 2 x  cosh x  sinh x
2
2
Notations
11 The inverse function of a hyperbolic function is an
inverse hyperbolic function. They are


inverse hyperbolic sine: sinh 1 x  ln x  x 2  1 , x  R,


inverse hyperbolic cosine: cos h 1 x  ln x  x 2  1 ,
x  [1, ),inverse hyperbolic tangent: tan h 1 x 
1 1 x
ln
, x   1,1
2 1 x
7、 Implicit functions
In applied problems we often need to investigate a class
of functions in which the correspondence rule between
the dependent variable y and the independent variable
x is defined by an equation of the form F  x, y   0,
when F  x, y  denotes an expression in x and y.
Definition 7
If there is a function y  f  x  , which is defined on
some interval I , such that F  x, f  x    0, x  I , then
y  f  x  , x  I is called a implicit function defined by
the equation F  x, y   0.