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【托马斯微积分】（12版）阅读笔记1：函数_a simple example of a rational function is the fun
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一.words
d
o
m
a
i
n
domain
domain 定义域
n
a
t
u
r
a
l
natural
natural
d
o
m
a
i
n
domain
domain 自然定义域
r
a
n
g
e
range
range 值域
v
a
r
i
a
b
l
e
variable
variable
q
u
a
n
t
i
t
y
quantity
quantity 变量
i
n
d
e
p
e
n
d
e
n
t
independent
independent
v
a
r
i
a
b
l
e
variable
variable 自变量
d
e
p
e
n
d
e
n
t
dependent
dependent
v
a
r
i
a
b
l
e
variable
variable 因变量
f
o
r
m
u
l
a
formula
formula 公式
r
a
d
i
u
s
radius
radius 半径
s
t
a
t
e
state
state 规定
p
o
s
i
t
i
v
e
positive
positive 正
n
e
g
a
t
i
v
e
negative
negative 负
n
o
n
n
e
g
a
t
i
v
e
nonnegative
nonnegative 非负
n
o
n
p
o
s
i
t
i
v
e
nonpositive
nonpositive 非正
r
e
a
l
v
a
l
u
e
d
realvalued
realvalued
f
u
n
c
t
i
o
n
function
function 实值函数
i
n
t
e
r
v
a
l
interval
interval 区间
o
p
e
n
open
open
i
n
t
e
r
v
a
l
interval
interval 开区间
c
l
o
s
e
d
closed
closed
i
n
t
e
r
v
a
l
interval
interval 闭区间
h
a
l
f
half
half
o
p
e
n
open
open
i
n
t
e
r
v
a
l
interval
interval 半开半闭区间
a
r
r
o
w
arrow
arrow
d
i
a
g
r
a
m
diagram
diagram 矢量图
v
e
r
i
f
y
verify
verify 检验
s
q
u
a
r
e
square
square 平方
s
q
u
a
r
e
square
square
r
o
o
t
root
root 平方根
a
r
i
t
h
m
e
t
i
c
arithmetic
arithmetic 算术，算法
r
e
c
i
p
r
o
c
a
l
s
reciprocals
reciprocals 倒数
C
a
r
t
e
s
i
a
n
Cartesian
Cartesian
p
l
a
n
e
plane
plane 笛卡尔平面（平面直角坐标系)
c
o
o
r
d
i
n
a
t
e
s
coordinates
coordinates 坐标
s
c
a
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t
e
r
p
l
o
t
scatterplot
scatterplot 散点图
T
h
e
The
The
V
e
r
t
i
c
a
l
Vertical
Vertical
L
i
n
e
Line
Line
T
e
s
t
Test
Test
f
o
r
for
for
a
a
a
F
u
n
c
t
i
o
n
Function
Function 函数的垂直性检验
P
i
e
c
e
w
i
s
e
−
D
e
f
i
n
e
d
Piecewise-Defined
Piecewise−Defined
F
u
n
c
t
i
o
n
Function
Function 分段倒数
a
b
s
o
l
u
t
e
absolute
absolute
v
a
l
u
e
value
value 绝对值
i
n
c
r
e
a
s
i
n
g
increasing
increasing
f
u
n
c
t
i
o
n
function
function 增函数
d
e
c
r
e
a
s
i
n
g
decreasing
decreasing
f
u
n
c
t
i
o
n
function
function 减函数
e
v
e
n
even
even
f
u
n
c
t
i
o
n
function
function 偶函数
o
d
d
odd
odd
f
u
n
c
t
i
o
n
function
function 奇函数
s
y
m
m
e
t
r
y
symmetry
symmetry 对称性
s
y
m
m
e
t
r
i
c
symmetric
symmetric 对称的
a
x
i
s
axis
axis 轴
l
i
n
e
a
r
linear
linear
f
u
n
c
t
i
o
n
function
function 一次函数
i
d
e
n
t
i
t
y
identity
identity
f
u
n
c
t
i
o
n
function
function 恒等函数
c
o
s
t
a
n
t
costant
costant
f
u
n
c
t
i
o
n
function
function常数函数
p
o
w
e
r
power
power
f
u
n
c
t
i
o
n
function
function幂函数
r
a
t
i
o
n
a
l
rational
rational
f
u
n
c
t
i
o
n
function
function 有理函数
a
l
g
e
b
r
a
i
c
algebraic
algebraic
f
u
n
c
t
i
o
n
function
function 代数函数
t
r
i
g
o
n
o
m
e
t
r
i
c
trigonometric
trigonometric
f
u
n
c
t
i
o
n
function
function 三角函数
e
x
p
o
n
e
n
t
i
a
l
exponential
exponential
f
u
n
c
t
i
o
n
function
function 指数函数
l
o
g
a
r
i
t
h
m
i
c
logarithmic
logarithmic
f
u
n
c
t
i
o
n
function
function 对数函数
i
n
v
e
r
s
e
inverse
inverse
f
u
n
c
t
i
o
n
function
function 反函数
t
r
a
n
s
c
e
n
d
e
n
t
a
l
transcendental
transcendental
f
u
n
c
t
i
o
n
function
function 超越函数
h
y
p
e
r
b
o
l
a
hyperbola
hyperbola 双曲线
s
l
o
p
e
slope
slope 斜率
o
r
i
g
i
n
origin
origin 原点
p
r
o
p
o
r
t
i
o
n
a
l
proportional
proportional 成比例
m
u
l
t
i
p
l
e
multiple
multiple 倍数
二 .
1.Functions; Domain and Range
y
=
f
(
x
)
y=f(x)
y=f(x) (“y equals f of x”)
A function ƒ from a set D to a set Y is a rule that assigns a unique (single) element
f
(
x
)
∈
Y
f(x)\in Y
f(x)∈Y to each element
x
∈
D
x \in D
x∈D.
Notice that a function can have the same value at two different input elements in the domain, but each input element x is assigned a single output value ƒ(x).
2.Graphs of Functions
If ƒ is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for ƒ. In set notation, the graph is { ( x , f ( x ) ) ∣ x ∈ D } \{(x,f(x))|x\in D\} {(x,f(x))∣x∈D}
3.Representing a Function Numerically
The graph consisting of only the points in the table is called a scatterplot.
4.The Vertical Line Test for a Function
A function ƒ can have only one value for each x in its domain, so no verticalline can intersect the graph of a function more than once. If a is in the domain of the function ƒ, then the vertical line x = a x=a x=a will intersect the graph of ƒ at the single point ( a , f ( a ) ) (a,f(a)) (a,f(a)) .
5.Piecewise-Defined Functions
Sometimes a function is described by using different formulas on different parts of its domain. One example is the absolute value function
The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function or the integer floor function. It is denoted
⌊
x
⌋
⌊x⌋
⌊x⌋ .
The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function or the integer ceiling function. It is denoted
⌈
x
⌉
⌈x⌉
⌈x⌉ .
6.Increasing and Decreasing Functions
7.Even Functions and Odd Functions: Symmetry
8.Common Functions
8.1 Linear Functions
A function of the form
ƒ
(
x
)
=
m
x
+
b
ƒ(x)=mx+b
ƒ(x)=mx+b, for constants m and b, is
called a linear function.
The function
ƒ
(
x
)
=
x
ƒ(x)=x
ƒ(x)=x where
m
=
1
m = 1
m=1 and
b
=
0
b = 0
b=0 is called the identity function.
Constant functions result when the slope m = 0
A linear function with positive slope whose graph passes through the origin is called a proportionality relationship.
Two variables y and x are proportional (to one another) if one is always a constant multiple of the other; that is, if y = k x y = kx y=kx for some nonzero constant k.
If the variable y is proportional to the reciprocal 1 / x 1/x 1/x, then sometimes it is said that y is inversely proportional to x (because 1 / x 1/x 1/x is the multiplicative inverse of x).
8.2 Power Functions
A function f ( x ) = x a f(x)=x^a f(x)=xawhere a is a constant, is called a power function.
8.3 Polynomials
A function p is a polynomial if p ( x ) = a n x n + a n − 1 x n − 1 + . . . + a 1 x + a 0 p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0 p(x)=anxn+an−1xn−1+...+a1x+a0 where n is a nonnegative integer and the numbers a 0 , a 1 , a 2 . . . a n − 1 , a n a_0,a_1,a_2...a_{n-1},a_n a0,a1,a2...an−1,anare real constants(called the coefficients of the polynomial). If the leading coefficient an a n ! = 0 a_n!=0 an!=0 and n > 0 n>0 n>0, then n is called the degree of the polynomial.
8.4 Rational Functions
A rational function is a quotient or ratio ƒ ( x ) = p ( x ) / q ( x ) ƒ(x) = p(x)/q(x) ƒ(x)=p(x)/q(x), where p and q are polynomials. The domain of a rational function is the set of all real x for which q ( x ) ! = 0 q(x)!=0 q(x)!=0
8.5 Algebraic Functions
Any function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots) lies within the class of algebraic functions. All rational functions are algebraic, but also included are more complicated functions.
8.6 Trigonometric Functions
8.7 Exponential Functions
Functions of the form f ( x ) = a x f(x)=a^x f(x)=ax , where the base a > 0 a > 0 a>0 is a positive constant and a ! = 0 a!=0 a!=0 have domain are called exponential functions.
8.8 Logarithmic Functions
These are the functions ƒ ( x ) = l o g a x ƒ(x) = log_a^x ƒ(x)=logax, where the base a ! = 1 a!=1 a!=1 is a positive constant. They are the inverse functions of the exponential functions
8.9 Transcendental Functions
These are functions that are not algebraic. They include the trigonometric, inverse trigonometric, exponential, and logarithmic functions, and many
other functions as well. A particular example of a transcendental function is a catenary.
