Outline
Chapter One
Functions and Models
Section 1.1
Section 1.1
Four Ways to Represent a
Function
Four Ways (cont’d)
z
z
Goals
z
Learn to represent functions using
z
Words（文字敘述）
z Tables of values（表格）
z Graphs（圖形）
z Formulas（公式）
z
z
Present the Vertical Line Test for curves in the
plane （判斷函數的方法）
What is a Function?
z
z
Discuss piecewise defined functions（分段定義函
數）, including the absolute value function
Study symmetry of functions（函數的對稱性）,
including even and odd functions
Discuss increasing and decreasing functions（遞
增或遞減函數）
A function arises whenever one quantity（量）
depends on another.（當一個量與另一個量
有關時，函數就自然的產生或定義了）
What is a Function?
z
Here are four examples:
A.
B.
C.
The area A of a circle depends on the radius r
of the circle. （圓面積與半徑有關）
z A and r are connected by the rule A = πr2 .
The population（人口） P of the world depends
on the time t , even though the value of P(t)
can only be estimated. （人口數與時間有關）
The cost C of mailing a first-class letter
depends on the weight w of the letter. （信件
的郵資與時重量有關）
z There is no simple formula connecting w and
C, but the post office can find C given w .
1
What is a Function? (cont’d)
D.
The vertical acceleration a of the ground
during an earthquake（地震） is a function of the
elapsed time t .（不同時間，地震的垂直加速度
就不同。地震的垂直加速度與時間有關）
z
Seismic（地震的） activity during the Northridge
earthquake that shook Los Angeles in 1994.
In General…
z
A function is a rule（規則） that assigns（指定）…
to each element x in a set A …
exactly one element, called f(x) , in a set B .
函數就是某一種規則，它將集合A中的每一個元素x，指
定到集合B中的 一個（而且只有一個）元素，記為f(x)
z
z
z
Usually we will take A and B to be sets of real
numbers.
通常A與B為實數集合，但並不一定。
Terminology (cont’d)
Terminology
z
z
z
The set A is called the domain（定義域） of the
function.
The number f(x) is the value of f at x （在x這點的
函數值） and is read “ f of x ” .
The range（值域） of f is the set of all possible
values of f(x) as x varies throughout the
domain.
z
A symbol that represents an arbitrary number
in the domain of a function f is called an
independent variable.（表示定義域中數字的符號稱為自變
z
A symbol that represents a number in the
range of f is called a dependent variable. （表
數）
示值域中數字的符號稱為因變數）
z
In the function A = πr2 giving the area of a circle
in terms of its radius,
z
z
Diagram
r is the independent variable, and
A is the dependent variable.
Graph of a Function
z
z
z
The graph of a function f is the set of ordered
pairs {( x , f ( x ) )| x ∈ A} .
So the graph consists of all points (x, y) where
y = f(x) and x is in the domain of f .
From the graph of f we can read…
z
z
the value of f(x) as being the height of the graph
above the point x ;
the domain and range of f , as the next slide shows:
2
Graph (cont’d)
Example
The graph of a function f
is shown at right.
z
a)
b)
Example (cont’d)
z
Example
Solution
a)
b)
Find the values of f(1)
and f(5) .
What are the domain and
range of f ?
z
f(1) = 3 . f(5) ≈ – 0.7 .
the domain of f is the closed interval [0, 7] .
the range of f is [– 2, 4] .
Sketch the graph and find the domain and
range of each function:
a) f(x) = 2x – 1
z
b) g(x) = x2
Solution (See figures on the next slide.)
a)
b)
The graph of f is a line with slope 2 and
y-intercept –1 , so the domain and range of f
are both R .
The graph of the equation y = x2 is a parabola,
as shown. The domain of g is R , but the
range of g is [0, ∞) .
Example
Example (cont’d)
z
Discuss the domain and range of
(a ). f ( x) =
1
x2 −1
x−2
(b). f ( x) = 2
x − x − 12
(c). f ( x) = 1 − x
3
Representations of Functions
z
z
z
z
Let’s revisit the four sample functions at the
beginning of this section:
z
Functions can be represented in four ways:
z
z
Representations (cont’d)
Verbally (that is, by a description in words)
Numerically (by a table of values)
Visually (by a graph)
Algebraically (by an explicit formula)
The area of a circle as a function of its radius
was given algebraically.
The world population as a function of time was
given verbally.
A.
B.
It is often useful to convert from one
representation to another, where possible.
Representations (cont’d)
Representations (cont’d)
The cost of mailing a letter as a function of weight
was also given verbally, but a table of values could
be given as well:
C.
D.
The vertical acceleration of the ground in an
earthquake, as a function of time, was given
graphically.
z
z
Example (cont’d)
Example
z
z
A rectangular storage container with an open
top has a volume of 10 m3 .
The length of its base is twice its width, and
material for the…
z
z
z
base costs $10 per square meter;
sides costs $6 per square meter.
Express the cost of materials as a function of
the width of the base.（將材料的成本表成基底寬度的函
Other representations for this function could be
given…
but the graphical one is the most convenient in this
case.
z
Solution We add the costs of
the base and four sides.
z
At right are shown the width w
and height h .
z
The area of the base is (2w)w
= 2w2 , so the material for the
base costs 10(2w2) .
z
Note that the length is 2w .
數）
4
Graphs of Functions
Example (cont’d)
z
The cost of material for the sides is
6[2(wh) + 2(2wh)] , so the total cost is
C = 10(2w2) + 6[2(wh) + 2(2wh)] = 20w2 + 36wh .
z
z
Next, since the volume is 10 m3 , w(2w)h = 10 ,
so that h =
z
z
10
5
=
.
2w2 w 2
z
The graph of a function is a curve in the xyplane, but…
Which curves in the xy-plane are graphs of a
function?
The answer is given by the Vertical Line Test:
Substituting then gives C as a function of w :
5
180
C ( w ) = 20w 2 + 36w ⎛⎜ 2 ⎞⎟ = 20w 2 +
, w >0.
w
⎝w ⎠
Graphs of Functions (cont’d)
Piecewise Defined Functions
z
z
Sometimes functions are defined by different
formulas in different parts of their domains.
⎧1 − x if x ≤ 1
, then...
Example: If f ( x ) = ⎨ 2
if x > 1
⎩ x
z
z
z
Piecewise (cont’d)
z
To graph f , note that…
z
z
z
for x ≤ 1 , the graph of f
must coincide with the line
y = 1 – x , whereas
for x > 1 , the graph must
coincide with the parabola
y = x2 .
since 0 ≤ 1 , we have f(0) = 1 – 0 = 1 ;
since 1 ≤ 1 , we have f(1) = 1 – 1 = 0 ;
since 2 > 1 , we have f(2) = 22 = 4 .
The Absolute Value Function
z
z
This is an important example of a piecewise
defined function.
First, recall the definition of the absolute value
of a number a :
Here is the graph:
5
Absolute Value (cont’d)
z
To graph f(x) = |x| , note
that…
z
z
If f(– x) = f(x) for every number x in its domain,
then f is called an even function.（偶函數）
z
z
to the right of the y-axis the
graph of f must coincide
with the line
y = x , whereas
to the left of the y-axis the
graph of f must coincide
with the line y = – x :
Even and Odd (cont’d)
z
Even and Odd Functions
If f(– x) = – f(x) for every
number x in its domain,
then f is called an odd
function.（奇函數）
f(x) = x3 is odd because
f(– x) = (– x)3 = – x3 = – f(x) .
z 把y(x)軸的另一邊，對稱至x
(y)軸，會與原先的另一邊對
稱於y(x)軸。
z
f(x) = x2 is even because
f(– x) = (– x)2 = x2 = f(x) .
圖形為對稱於y軸。
Three Examples
a)
f(x) = x5 + x is an odd function, since
f(– x) = (– x)5 + (– x) = – (x5 + x) = – f(x) .
b)
g(x) = 1 – x4 is an even function, since
g(– x) = 1 – (– x)4 = 1 – x4 = g(x) .
z
Examples (cont’d)
c)
However h(x) = 2x – x2 is neither, since
h(– x) = 2(– x) – (– x)2 = – 2x – x2 .
Increasing and Decreasing
Functions
z
The graph shown
z
z
z
rises from A to B , (increasing, 遞增 )
falls from B to C , (decreasing, 遞減 )
and then rises again from C to D .
6
Increasing and Decreasing
Functions
z
Example
In general we say that a function f is…
z
z
z
increasing on an interval I if f(x1) < f(x2)
whenever x1 < x2 in I , and
decreasing on I if f(x1) > f(x2) whenever
x1 < x2 in I .
z
z
Four ways to represent functions
z
z
z
z
z
z
Verbal
Numeric
Visual
Algebraic
decreasing on the
interval (– ∞, 0] , and
increasing on the
interval [0, ∞) .
Review (cont’d)
Review
z
As the graph shows,
f(x) = x2 is…
z
Symmetry of graphs
z
Increasing and decreasing functions
z
Even and odd functions
Vertical Line Test for curves in the plane
Piecewise defined functions
z
Absolute value function
Section 1.2
Section 1.2
A Catalog of Essential Functions
（重要函數的總集）
A Catalog of Essential Functions
（重要函數的總集） (cont’d)
z
Goals
z
Discuss the following types of function used in
mathematical models:
z
z
Linear
Polynomial
Polynomial
Power and root
z Rational
z Algebraic
z Trigonometric
z Exponential and logarithmic
z Transcendental
z
z
7
Linear Functions
z
These are of the general form
Polynomials（多項式）
z
y = f(x) = mx + b ,
z
Thus, the linear function f(x) = 3x – 2 grows three
times as fast as x does.
Special Cases
z
z
z
z
z
z
n is a nonnegative integer, and
the numbers a0 , a1 , a2 , … , an are constants
called the coefficients （係數）of the polynomial.
If an ≠ 0 , then n is called the degree （階）
of the polynomial.
Sample Quadratic Functions
1 is of the form P(x) = mx + b , and so is a linear
function（線性函數）;
2 is of the form P(x) = ax2 + bx + c , and is called
a quadratic function（二次函數）. Its graph is
always a parabola （拋物線）(illustrated on the
next slide);
3 is of the form P(x) = ax3 + bx2 + cx + d , and is
called a cubic function（三次函數）.
Power Functions（冪函數）
z
where…
A polynomial of degree…
z
z
P(x) = anxn + an-1xn-1 + ∙∙∙ + a1x + a0 ,
where m is the slope（斜率） and b is the
y-intercept（截距）.
Notice that linear functions grow at a constant
rate.（遞增或遞減的速度是固定的）
z
A polynomial is a function of the form
Graphs of f(x) = xn , n = 1, 2, 3, 4, 5
A power function is of the form f(x) = xa , where
a is a constant.
We consider several cases:
z
If a = n , where n is a positive integer, then…
the general shape of the graph depends on whether n
is even or odd.
z The next slide shows the special cases n = 1, … ,5 .
z The following slide illustrates the general situation.
z
8
Families of Power Functions
z
z
Power Functions (cont’d)
z
When n is even, f(x) = xn is an even function;
When n is odd, f(x) = xn is an odd function:
If a = 1/n , where n is a positive integer, then we
call f(x) = x1/n a root function. When n is…
f ( x ) = x1/ n = n x resembles
that of the square root function f ( x ) = x ;
z
even, the graph of
z
odd, the graph of f(x) = x1/n resembles that of the
cube root function f ( x ) =
Power Functions (cont’d)
z
A rational function is a ratio of two polynomials:
P ( x)
f ( x) =
,
Q( x)
z
z
x:
Power Functions (cont’d)
z
Rational Functions（有理函數）
3
If a = –1 , then f(x) = 1/x ,
with graph as shown.
Examples
z
The domain of f is the set
of all x ≠ ±2 .
where P and Q are polynomials.
The domain of f consists of all values of x
such that Q(x) ≠ 0 .
The reciprocal function f(x) = 1/x is a simple
example.
z
The domain of f is the set of all x ≠ 0 .
9
Algebraic Functions (cont’d)
Algebraic Functions（代數函數）
z
A function f is called algebraic if it can be
constructed using algebraic operations, such
as…
z
z
z
z
Two remarks:
z
z
Addition/subtraction
Multiplication/division
Taking roots
g ( x) =
…starting with polynomials.
多項式經由加檢乘除、開根號等代數運算所得的
函數，統稱為代數函數。
Algebraic Functions (cont’d)
z
Here are some further examples of algebraic
functions, together with their graphs:
Any rational function is automatically an algebraic
function.
However, many algebraic functions are not rational,
for example
有理函數沒有開根號。
Trigonometric Functions
（三角函數）
z
The two best-known trigonometric functions
(graphed on the next slide) are the…
z
z
z
z
Graphs of sin x and cos x
x 4 − 16 x 2
+ ( x − 2) 3 x + 1 .
x+ x
sine function, denoted by sin x , and the
cosine function, denoted by cos x .
Both sin x and cos x always lie between –
1 and 1 , regardless of x .
Also, sin x = 0 when x = nπ , where n is
an integer.
Trigonometric Functions (cont’d)
z
z
An important property of the sine and cosine
functions is that they are periodic with period
2π .
This means that for all values of x ,
sin (x + 2π) = sin x and
cos (x + 2π) = cos x
z
.
This makes the sine and cosine functions
suitable for modeling repetitive phenomena.
10
Exponential Functions
指數函數
Trigonometric Functions (cont’d)
z
The tangent function is
given by
tan x =
z
sin x
,
cos x
z
Exponential functions are of the form
f(x) = ax , where the base a is a positive
constant.
注意與與冪函數不同。
and is undefined
whenever cos x = 0 ,
that is, when
x = ±π/2, ±3π/2, … .
z
Exponential Functions
指數函數
z
Logarithmic Functions
對數函數
Two special cases a = 2 and a = 0.5 .
z
z
In both cases the domain is (–∞, ∞) , and the
range is (0, ∞) .
a>1,
increasing
Four logarithmic
functions with various
bases.
z
z
In each case the
domain is (0, ∞) and
the range is (–∞, ∞) .
與指數函數相反。
The logarithmic functions f(x) = loga x ,
where the base a is a positive constant,
are the inverse functions （反函數）of the
exponential functions.
a<1,
decreasing
Transcendental Functions
超越函數
Logarithmic Functions
對數函數
z
f(x) = xa 。
z
z
These are functions that are not algebraic.
The set of transcendental functions includes
the…
z
z
z
trigonometric/inverse trigonometric and
exponential/logarithmic functions, but…
it also includes a vast number of other functions
that have never been named!
11
Section 1.3
Review
z
One point in this section:
z
The various types of mathematical function used in
modeling.
New Functions From Old
Functions（函數的轉換）
z
Goals
z
Learn to create new functions from existing ones
by…
Translations, consisting of vertical and horizontal
shifting
z Vertical and horizontal stretching and reflecting
z Algebraic combinations
z Composition
z
Translation
z
z
Translation
z
To translate or shift （平移）a graph is to move it
up, down, left, or right without changing its
shape.
函數g是將f的圖形，上移、下移、左移、右移？
z
z
上下移（x固定，y變動）
左右移（y固定，x變動）
Translation is summarized by the following
table and illustration:
f ( x) = x 2
g1 ( x) = x 2 + 3
g 2 ( x) = x 2 − 3
g 3 ( x) = ( x − 3)
2
g 4 ( x) = (x + 3)
2
Stretching and Reflecting
Translation (cont’d)
z
放大與縮小
z
z
z
水平，x軸（ y固定，x變動）
垂直，y軸（ x固定，y變動）
鏡射
z
x軸，y軸。
f ( x) = x 2
g1 ( x) = 3 x 2
g 2 ( x) =
g 3 ( x) = (3 x )
2
1 2
x
3
⎛1 ⎞
g 4 ( x) = ⎜ x ⎟
⎝3 ⎠
2
g5 ( x) = − x 2
12
Stretching and Reflecting (cont’d)
z
Stretching and Reflecting (cont’d)
Stretching and reflecting are summarized by the
following table and illustration:
Example (cont’d)
Example
z
z
z
Figure 3 illustrates these stretching
transformations when applied to the cosine
function with c = 2 .
For instance, to get the graph of y = 2 cos x
we multiply the y-coordinate of each point on
the graph of y = cos x by 2 .
This means that the graph of y = cos x gets
stretched vertically by a factor of 2 .
Example
z
Example
Sketch the graph of the function
f(x) =
x2
Solution First we complete the square:
y = x2 + 6x + 10 = (x + 3)2 + 1 .
+ 6x + 10 .
請用 y = x2 的圖形，經由各種平移(translation)得到
上述函數的圖形。
z
z
Thus we get the graph of f by starting with
the parabola y = x2 and shifting…
3 units to the left （往左平移三單位）, and
1 unit upward, （往上平移一單位）
as shown on the next slide:
z
z
13
Example (cont’d)
Example
z
Sketch the graphs of
a)
b)
y = sin 2x
y = 1 – sin x
請用 y = sin x 的圖形，經由各種平移(translation)、
放大縮小(stretch)及鏡射(reflect)得到上述函數的
圖形。
Example
z
Sketch the graphs of
a)
b)
y = sin 2x
y = 1 – sin x
Example (cont’d)
z
Solution
a)
We get the graph of y = sin 2x from that of y =
sin x by compressing horizontally by a factor of
2 .（橫軸方向擠壓兩單位）
z
b)
To get the second graph we first…
z
z
Example (cont’d)
Thus, whereas the period of y = sin x is 2π , the
period of y = sin 2x is 2π/2 = π :
reflect the graph of y = sin x about the x-axis to
get the graph of y = – sin x , and then
shift 1 unit upward to get y = 1 – sin x :
Example (cont’d)
14
Combinations (cont’d)
Combinations of Functions
z
Two functions f and g can be combined to
form new functions
z
z
z
z
z
This is summarized in the following table:
f+g,
f–g,
f g , and
f/g
just as we add, subtract, multiply, and divide
real numbers.
Example (cont’d)
Example
z
z
If f ( x ) =
x and g ( x ) = 4 − x 2 , find the
z
Here are the formulas for the functions we
want:
functions f + g , f – g , f g , and f/g .
( f + g ) ( x ) = x + 4 − x2
Solution The intersection of the domains of f
and g is [0, ∞) ∩ [– 2 , 2] = [0, 2] .
( f − g ) ( x ) = x − 4 − x2
z
This is therefore the domain of f + g , f – g , and
fg.
z
The domain of f/g , however, is [0, 2) , since g(2)
=0.
Composition（合成） of Functions
z
z
This is another way of combining functions.
As an example, suppose that y = f ( u ) = u
and u = g(x) = x2 + 1 :
z
z
z
( fg ) ( x ) = x 4 − x2 = 4 x − x 3
⎛f⎞
x
x
=
⎜ g ⎟( x) =
2
−
4
x2
−
4
x
⎝ ⎠
Composition (cont’d)
z
z
We say the new function is the composition
（合成） of f and g .
Here is the general definition:
Since y is a function of u , and…
u is in turn a function of x , it follows that…
y is ultimately a function of x :
y = f ( u ) = f ( g ( x ) ) = f ( x 2 + 1) = x 2 + 1 .
z
Next we illustrate this using a machine and
an arrow diagram:
15
Two Diagrams (cont’d)
Two Diagrams
z
In this machine diagram, the f ◦ g machine is
composed of the…
z
z
z
g machine (first) and then
the f machine:
Example
z
z
Here is the corresponding arrow diagram for f ◦
g:
Example
If f(x) = x2 and g(x) = x – 3 , find the
composite functions f ◦ g and g ◦ f .
Solution By definition of composition,
(f ◦ g)(x) = f(g(x)) = f(x – 3) = (x – 3)2
and
z
If f ( x ) = x and g ( x ) = 2 − x ,
then find each function and its domain:
a)
b)
c)
d)
f◦g
g◦f
f◦f
g◦g
(g ◦ f)(x) = g(f(x)) = g(x2) = x2 – 3 .
z
Note that in general f ◦ g ≠ g ◦ f !
Example (cont’d)
z
Example (cont’d)
Solution
a)
First we have
( f o g )( x) = f ( g ( x)) =
c)
f ( 2 − x) =
d)
{x|2 − x ≥ 0} = {x| x ≤ 2} = ( −∞ ,2 ] .
b)
Next
( g o f )( x) = g ( f ( x)) = g (
( f o f )( x ) = f ( f ( x)) =
2−x = 4 2−x .
The domain of f ◦ g is
x) = 2 − x .
Then
f ( x) =
x=4x,
and the domain of f ◦ f is [0, ∞) .
Finally
( g o g )( x) = g ( g ( x)) = g (
2 − x) = 2 − 2 − x .
This expression is defined when x ≤ 2 and
2 − 2 − x ≥ 0 , which is equivalent to the closed
interval [– 2, 2] .
The domain of g ◦ f is the interval [0, 4] .
16
Example
Example
Answer
Example (cont’d)
Example
Example (cont’d)
Example
17
Example (cont’d)
Example
Example (cont’d)
Example
Review
z
We can combine old functions to obtain new
ones in several ways:
z
z
z
z
Translations
Stretching and reflecting
Algebraic operations
Composition
18