【托马斯微积分】（12版）阅读笔记1：函数

一.words

domaindomaindomain 定义域 naturalnaturalnatural domaindomaindomain 自然定义域
rangerangerange 值域 variablevariablevariable quantityquantityquantity 变量
independentindependentindependent variablevariablevariable 自变量 dependentdependentdependent variablevariablevariable 因变量
formulaformulaformula 公式 radiusradiusradius 半径
statestatestate 规定 positivepositivepositive 正
negativenegativenegative 负 nonnegativenonnegativenonnegative 非负
nonpositivenonpositivenonpositive 非正 realvaluedrealvaluedrealvalued functionfunctionfunction 实值函数
intervalintervalinterval 区间 openopenopen intervalintervalinterval 开区间
closedclosedclosed intervalintervalinterval 闭区间 halfhalfhalf openopenopen intervalintervalinterval 半开半闭区间
arrowarrowarrow diagramdiagramdiagram 矢量图 verifyverifyverify 检验
squaresquaresquare 平方 squaresquaresquare rootrootroot 平方根
arithmeticarithmeticarithmetic 算术，算法 reciprocalsreciprocalsreciprocals 倒数
CartesianCartesianCartesian planeplaneplane 笛卡尔平面（平面直角坐标系) coordinatescoordinatescoordinates 坐标
scatterplotscatterplotscatterplot 散点图 TheTheThe VerticalVerticalVertical LineLineLine TestTestTest forforfor aaa FunctionFunctionFunction 函数的垂直性检验
Piecewise−DefinedPiecewise-DefinedPiecewise−Defined FunctionFunctionFunction 分段倒数 absoluteabsoluteabsolute valuevaluevalue 绝对值
increasingincreasingincreasing functionfunctionfunction 增函数 decreasingdecreasingdecreasing functionfunctionfunction 减函数
eveneveneven functionfunctionfunction 偶函数 oddoddodd functionfunctionfunction 奇函数
symmetrysymmetrysymmetry 对称性 symmetricsymmetricsymmetric 对称的
axisaxisaxis 轴 linearlinearlinear functionfunctionfunction 一次函数
identityidentityidentity functionfunctionfunction 恒等函数 costantcostantcostant functionfunctionfunction常数函数
powerpowerpower functionfunctionfunction幂函数 rationalrationalrational functionfunctionfunction 有理函数
algebraicalgebraicalgebraic functionfunctionfunction 代数函数 trigonometrictrigonometrictrigonometric functionfunctionfunction 三角函数
exponentialexponentialexponential functionfunctionfunction 指数函数 logarithmiclogarithmiclogarithmic functionfunctionfunction 对数函数
inverseinverseinverse functionfunctionfunction 反函数 transcendentaltranscendentaltranscendental functionfunctionfunction 超越函数
hyperbolahyperbolahyperbola 双曲线 slopeslopeslope 斜率
originoriginorigin 原点 proportionalproportionalproportional 成比例
multiplemultiplemultiple 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)∈Yf(x)\in Yf(x)∈Y to each element x∈Dx \in Dx∈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=ax=ax=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)=mx+bƒ(x)=mx+bƒ(x)=mx+b, for constants m and b, is
called a linear function.
The function ƒ(x)=xƒ(x)=xƒ(x)=x where m=1m = 1m=1 and b=0b = 0b=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=kxy = kxy=kx for some nonzero constant k.

If the variable y is proportional to the reciprocal 1/x1/x1/x, then sometimes it is said that y is inversely proportional to x (because 1/x1/x1/x is the multiplicative inverse of x).

8.2 Power Functions

A function f(x)=xaf(x)=x^af(x)=xawhere a is a constant, is called a power function.

8.3 Polynomials

A function p is a polynomial ifp(x)=anxn+an−1xn−1+...+a1x+a0p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0p(x)=anxn+an−1xn−1+...+a1x+a0 where n is a nonnegative integer and the numbers a0,a1,a2...an−1,ana_0,a_1,a_2...a_{n-1},a_na0,a1,a2...an−1,anare real constants(called the coefficients of the polynomial). If the leading coefficient an an!=0a_n!=0an!=0 and n>0n>0n>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)!=0q(x)!=0q(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)=axf(x)=a^xf(x)=ax , where the base a>0a > 0a>0 is a positive constant and a!=0a!=0a!=0 have domain are called exponential functions.

8.8 Logarithmic Functions

These are the functions ƒ(x)=logaxƒ(x) = log_a^xƒ(x)=logax, where the base a!=1a!=1a!=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.