Finite Interval（
有限區間
）open interval ( , )a b {x a x b} closed interval
a b, {x a x b}half open( or half closed) interval
( , ] { }
[ , ) { }
a b x a x b a b x a x b
§ 1.1 Functions and Their Representations 函數及描述方式
[ , )a b {x a x b} Infinite Interval（
無限區間
）[ , ) { }
( , ] { }
( , ) { }
( , ) { }
( , ) { }
a x a x
b x x b
a x a x
b x x b
x x
ℝ
: closed interval : closed interval : open interval : open interval
: both closed and open
Def
A function(
函數
) f from a set A to a set B is a rule that assigns to each element x in A exactly one (恰有一個
）element y, called f (x), in B
the domain( of , denoted as .
the range( of , denoted as
( ) ( ) .
f
f A f f
f D
x A f R
A
x
定義域）
值域）
each input x one and only one output y
[Ex] A { , , } a b c B {1, 2, 3, 4}
(a ) :
2
( ) ( )
( ) 1
3
f A B
a b c
f a f b
f c
f is a function
f is a one-to-one function
(b) :
1 3 3
g A B
a b c
g is a function
g is not a one-to-one function f is a one-to-one function
, , ( )
( ), ( ), ( ) 1, 2, 3
f
f
D A a b c
R f x x A
f a f b f c B
g is not a one-to-one function
, , ( )
( ), ( ), ( ) 1, 3
g
g
D A a b c
R g x x A
g a g b g c
（一對一函數）
[Ex]
(c) :
1
h A B
a
h is not a function
(not exactly one!) 2
(b) y2 x (x 0)
x x (a ) 1
:
y x f
x y
ℝ ℝ
f
f
D R
ℝ ℝ
f is a one-to-one function
(b) 2 ( 0) : 0,
y x x
f
x y
ℝ
f is not a function since
(not exactly one!)
Exercise:
:
f x y
Is f a function when
2
2
2 2
(a ) 1 ?
(b) 1 ?
(c) 1 ?
(d) 1 ?
x y
x y
x y
x y
[Sol:]
(a ) 1 1
:
x y y x
f x y
Since , there is exactly one value of y for each x, f is a function.
2 2
(b) 1 1
:
x y y x
f x y
1 2
y x
(c) x y2 1 y 1 x (d) x2 y2 1 y 1 x2 1
y x Since , there is
exactly one value of y for each x, f is a function.
(c) 2 1 1
:
x y y x
f x y
Since , there are more than one value of y, f is not a function.
1 y x
2 2 2
(d) 1 1
:
x y y x
f x y
1 2
y x
Since , there are more than one value of y,
f is not a function.
1 :
y x f
x y
ℝ ℝ
Since 1 , we have ( ) 1
or ( ) 1
y x y f x x
f x x
So, Df {x x ℝ} ℝ ( , )
(the set of all allowable values of x)
{ } ( , )
Rf y y ℝ
(the set of all possible function values f (x) )
[Ex] Find the domain(
定義域
) and range (值域
）of the function.{ 0}
( , 0) (0, ) {0}
(a ) ( ) 1
f
x
D x
f x
x
ℝ
{ 0}
( , 0) (0, ) {0}
Rf y y
ℝ
{ 0}
[0, )
(b)
{ 0}
[0 )
) , (
f
f
f x D
y x x x
R y
3
( , )
(c) ( )
( , )
f
f
D
R
f x x
ℝ
ℝ
(i) :
3 0 3
{ 3} [3, ) (ii) :
3 0
3 0 (d) ( ) 3
f
f
f
D
x x
D x x
R x
y x
x x
f
∵
(
{ 0}
[0, ) e) ( )
f
f
D
R y
f
y
x x
ℝ
3 0
f { 0} [0, )
y x
R y y
Exercise:
(a ) ( ) 1
1
(b) ( ) 2 1
(c) ( )
f x x
f x x
f x x
1. Find the domain（
定義域
） and range(值域
） of each of the following functions(c) ( )f x
x
2. ( ) 1 . Find
2 g
g x x D
x
[Sol]:
1. (a ) ( ) 1
1
1
0
f
f
f x x
D x x
R y y
(b) ( ) 2 1
2
1
f
f
f x x
D x x
R y y
(c) ( )
0
1,1
f
f
f x x
x
D x x
R
when 0, ( ) 1
when 0, ( ) 1
x f x x
x x f x x
x
2. ( ) x 1
g x
when x 0, ( )f x x 1
2. ( ) 1
2
2 0 2
1 0 1
1, 2
1, 2 2, 1, - 2
g
g x x
x
x x
x x
D x x x
The Graph of a Function（函數的圖形）
[Ex4] [Ex7] The piecewise defined function（分段定義函數） is given by . Sketch（畫圖）the graph.
The graph of a function f (x) is the graph of the equation i.e. the set
( ) y f x
x f x, ( ) x Df
.2
1 , if 1 ( ) , if 1
x x
f x x x
（方程式）
0,
f
f
D R
ℝ
The graph of the absolute value function（絕對值函數圖形）
[Ex] Graph the function y x2 4
(ii) then graph y x2 4 (i) first graph y x2 4
Exercise:
2
1 , 1
(3) Find and , where
, 1
f f
D R f x x
x x
(1) Graph the function using the skill you just learned.
(2) Graph the piecewise defined function（分段定義函數）:
y x
2
2, 1
( ) 1 , 1
x x
f x x x
x , x 1
[Sol]:
(1)
(2) ( ) x 2, x 1
f x
(3)
(2) ( ) 2, 2 1
1 , 1
x x
f x x x
2
(i)
(ii) when 1, 1 0 ie. ( ) 0 when 1, 1 ie. ( ) 1 So 0,
0
f
f
D
x x f x
x x f x
R
y y
ℝ (3)
Odd Function（奇函數）
Def: f (x) f x( ) f (x) f x( ) Graph: symmetric about the origin
Even Function（偶函數）
Def:
Graph: symmetric about the y-axis
（對稱） （對稱）
[Ex] f x( ) x3 [Ex]
3 3
Since ( ) ( )
( ) is an odd function
f x x x f x
f x
( ) 2
f x x
2 2
Since ( ) ( )
( ) is an even function
f x x x f x
f x
symmetric about the origin
symmetric about the y-axis
[Ex] Determine whether f is even, odd or neither(都不是).
[Sol]:
1 3
2
3
2
(a ) ( ) (b) ( ) 1 1 (c) ( ) 1
f x x f x
x f x x
x
1 1 3 3
2 2
(a ) ( ) ( ) ( ) is an odd function. ( ) ( )
1 1
(b) ( ) ( ) ( ) is an even function. ( ) ( )
1 1
f x x x f x f x f x f x
f x f x f x f x f x
x x
∵
∵
3 3
2 2
1 1
1 1
(c) ( )
Since ( ) ( ) and ( x x
f x x x
x x
f x f x f
)x f x( ). ( ) is neither even nor odd.f x
[Ex] A portion of the graph of a function f is given as follows.
(a) Complete the graph of f if it is even.
(b) Complete the graph of f if it is odd.
[Sol]:
Exercises:
(a) if f is even (b) if f is odd.
Exercises:
1. Determine whether f is even, odd or neither.
(a ) ( )f x x (b) ( )f x 3
2. The graph of a function f is given as follows. Is it even, odd or neither?
[Sol]:
1. (a) ( ) ( ) ( ) is even (b) ( ) 3 ( ) ( ) is even 2. (a) The graph is symmetric about the orign ( (b) The graph is not symmetric
f x x x f x f x
f x f x f x
f
∵
∵
∵
∵
對稱原點）
is odd
about the orign nor the y-axis (
f
不對稱原點與y軸）
is neither odd nor even is neither odd nor evenf
The Vertical Line Test（垂直線檢定）
A curve C in the xy-plane
C is the graph of a function of x
No vertical line intersects（相交）the curve more than once.
( )f a b
This is a graph of a function.
& ( )
a contradiction!
(
f a c
矛盾）
This is not a graph of a function.
§ 1.2 A Catalog of Essential Functions 常用基本函數概述
linear function
(1) : y mx b (its graph is a line 直線)
polynomial
(2) : p x( ) a xn n an1xn1 ⋯ a x2 2 a x1 a0 (an 0) : slope
: -intercept m
b y
斜率
截距
0 1 1
: nonnegative integer, called the degree .
, , , n , n : constants, called the coefficients . n
a a a a
⋯
次數
係數
（ ）
（ ）
(線性函數）
(多項式）
(i) A polynomial of degree 1 or 0 is a linear function
1 0
( ) or ( ) p x a x a p x a
(ii) A polynomial of degree 2 is a quadratic function（
二次多項式
）( ) 2 ( 0)
p x ax bx c a (its graph is a parabola) (iii) A polynomial of degree 3 is a cubic function（
三次多項式
）3 2
( ) ( 0)
p x ax bx cx d a
0 1 n1 n
(拋物線）
power function（冪函數）
(3) ： f x( ) xa (i) a= n ; n: nonnegative integer
p
( ) n : olynomial f x x
ex:
2 3
(a) ( )f x x (b) ( )f x x (c) ( )f x x
4 5
(d) ( )f x x (e) ( )f x x
(ii) when ; n: positive integer
（正整數）
ex. 1 1
3 3
(a) ( )f x x2 x (b) ( )f x x x a 1
n
1
root functio
( ) n n
f x x
（根式函數）
1 1
( )
(iii) 1. f x x reciprocal function
a x
（倒數函數）
rational function
(4) （有理函數）: ( ) ( ) ( ) f x P x
Q x
2
1 3 1 1
ex. ( ) , ( ) ( ) reciprocal function
2 4
x x
f x f x f x
x x x
P and Q are polynomials
algebraic function
(5) （代數函數）: a function which can be constructed using algebraic operations (such as addition（加）, subtraction（減）,
multiplication（乘）, division（除） and taking roots（取根式,開根號）) starting with polynomials.
x33
ex. ( ) 2 1 ( ) x
f x x f x
x x
transcendental function
(6) （超越函數）: the functions which are not algebraic.
ex.
1 1 1
: ( ) sin , cos , tan , : ( ) sin , trigonometric functions
inverse trigonometric functions exponential functions
logarithmic fu
cos , tan , :
ncti
( ) x ( 0, 1)
f x x x x
f x x x x
f x a a a
⋯
⋯
（三角函數）
（反三角函數）
（指數函數）
: ( ) log ( 0, ns
o （對數函數） f x a x a a 1)
[Ex5] Classify the function as one of the types of functions that we just discussed.
5
4
2
(a) ( ) 5 (b) ( ) (c) ( ) 1 1 (d) ( ) 1 5 (e) ( ) 3 (f) ( ) sin 5
2
x x
f x f x x f x
x
u t t t f x x f x x
x x
[Sol]:
5
(a) ( ) 5 : exponential function. (transcendental function) (b) ( ) : power function. (polynomial of degree 5)
f x x
f x x
4
(c) ( ) 1 : algebraic function.
1
(d) ( ) 1 5 : polynomial of degree 4.
(e
f x x
x
u t t t
2
) ( ) 3 : rational function.
2
(f) ( ) sin 5 : trigonometric function. (transcendental function) f x x
x x
f x x
(i) trigonometric function (Appendix (附錄) A) 2 rad 360 or rad 180
So. 1 rad 180 57.3 1 rad 0.017 rad.
180
[Ex] (a) Find the radian（弧度） measure of 60°
(b) Express in degrees.5 rad 4
[Sol]:
(a) 60 60 rad rad
(a) 60 60 rad rad
180 3
5 5 180
(b) rad 225
4 4
Degrees 0° 30° 45° 60° 90° 120° 135° 150° 180° 270 ° 360°
Radians 0 2 3 5 3 2
6 4 3 2 3 4 6 2
(hypotenuse)
(adjacent)
(opposite)
0 2
sin csc cos sec tan cot
b c
c b
a c
c a
b a
a b
(斜邊)
(對邊)
(鄰邊)
sin csc
cos sec tan cot
y r
r y
x r
r x
y x
x y
In calculus, we use radians
（弧度）
to measure angles（角度）
.
sin
c ,
,
os
sin
c ,
,
os
sin
c ,
,
os
sin
c ,
,
os
sine function
正弦函數
cosine function
餘弦函數
sin , , 1 1 1
cos , , 1 1 1
y x x y y
y x x y y
tan sin , 2 1 ,
cos 2
cot cos , , sin
y x x x n y
x
y x x x n y
x
tangent function
正切函數
cotangent function
餘切函數
sec 1 , 2 1 , 1 , 1 1
cos 2
csc 1 , , 1 , 1 1
y x x n y y y
x
y x x n y y y
secant function
正割函數
cosecant function
csc 1 , , 1 , 1 1
y x sin x n y y y
x
cosecant function
餘割函數
sin sin (odd func
sin 2 sin ( 2 )
tion
)
x x
x x
T
co co
s
s 2 cos ( 2 )
cos (even function ) x
x x
x T
tan
x
tan x cot
x
cot x
sec x 2 sec x csc
x 2
csc xDef ( informal )
We write if we can make the value of arbitrarily（任意地）
close to L ( as close to L as we like ) by taking x to be sufficiently close to a ( on either side of a ) but not equal to a （不等於a）
Notice : In finding limit of as x approaches a , we never consider x= a.
What matters is how f is defined near a . lim ( )
x a f x L
f
x
xf
§ 1.3 (2.2) The Limit of a Function 函數的極限
lim ( )
x a f x L
lim ( )
x a g x L
lim ( )
x ah x L
One – sided limit 單邊極限
the limit of ( ) as approaches lim ( )
the limit of ( ) as approaches from the left
x a
left hand f x x a
f x L
f x x a
Def ( informal )
We write
if we can make the value of arbitrarily close to L by taking x to be sufficiently close to a and less than a
Def ( informal )
We write
xf
the limit of ( ) as approaches
lim ( ) right hand f x x a
f x L
We write
if we can make the value of arbitrarily close to L by taking x to be sufficiently close to a and greater than a
xf
the limit of ( ) as approaches lim ( )
the limit of ( ) as approaches from the right
x a
right hand f x x a
f x L
f x x a
[Ex2] [Ex1] Guess the value of 2
21
1 1
lim let
1 1
x
x x
f x
x x
x 0.9 0.99 0.999 0.9999 → 1 ← 1.0001 1.001 1.01 1.1
f (x) 0.526 0.5025 0.50025 0.500025 0.499975 0.49975 0.4975 0.476
0.5
lim1 ( )
x f x
(Notice: does not exist)
0 . 5
lim
1 f x
x
1f
0 . 5
lim
1
x f
x
[Ex]
1
, 2
1
, 1 1
2
x x x
x x
g
li m1 ( ) 0 . 5
x g x
Notice:
1 2 exists
g
5 . 0 lim
5 . 0 lim
1 1
x g
x g
x x
[Ex3] [Ex2] Let , estimate
（估計）
the value oft -1.0 -0.5 -0.1 -0.05 -0.001 → 0 ← 0.001 0.05 0.1 0.5 1
f(t) 0.16558 0.16553 0.16662 0.16666 0.16667 0.16667 0.1666 0.16662 0.16553 0.16558
2 2 0
3 lim 9
t t
t
2
2
( ) t 9 3 f t
t
0.1666….
t 0
Notice : (1) 0 does not exist (2) lim 1
6 (3) lim 1
f
f t f t
6 1 3 lim
29
2
0
t
t
t
t 0
(3) lim
f t6
6
0 t
t
But if we take even smaller values of t and get the results from a calculator You can see something strange happens.
t -0.0005 -0.0001 -0.00005 -0.00001 0 0.00001 0.00005 0.0001 0.0005
0.16800 0.20000 0.00000 0.00000 0.00000 0.00000 0.20000 0.16800
( )
f tThe calculator gives false values !!
(代入更小的數值）
