一.words

$domain$?定義域??$natural$?$domain$?自然定義域
$range$?值域??$variable$?$quantity$?變量
$independent$?$variable$?自變量??$dependent$?$variable$?因變量
$formula$?公式??$radius$?半徑
$state$?規定??$positive$?正
$negative$?負??$nonnegative$?非負
$nonpositive$?非正??$realvalued$?$function$?實值函數
$interval$?區間??$open$?$interval$?開區間
$closed$?$interval$?閉區間??$half$?$open$?$interval$?半開半閉區間
$arrow$?$diagram$?矢量圖??$verify$?檢驗
$square$?平方??$square$?$root$?平方根
$arithmetic$?算術，算法??$reciprocals$?倒數
$Cartesian$?$plane$?笛卡爾平面（平面直角坐標系)??$coordinates$?坐標
$scatterplot$?散點圖??$The$?$Vertical$?$Line$?$Test$?$for$ ?$a$?$Function$?函數的垂直性檢驗
$Piecewise?Defined$?$Function$?分段倒數??$absolute$?$value$?絕對值
?$increasing$?$function$?增函數??$decreasing$?$function$?減函數
$even$?$function$?偶函數??$odd$?$function$?奇函數
$symmetry$? 對稱性??$symmetric$? 對稱的
$axis$? 軸??$linear$?$function$?一次函數
$identity$?$function$?恒等函數??$costant$?$function$常數函數
$power$?$function$冪函數??$rational$?$function$?有理函數
$algebraic$?$function$?代數函數??$trigonometric$?$function$?三角函數
$exponential$?$function$?指數函數??$logarithmic$?$function$?對數函數
$inverse$?$function$?反函數??$transcendental$?$function$?超越函數
$hyperbola$?雙曲線??$slope$?斜率
$origin$?原點??$proportional$?成比例
$multiple$?倍數

二 .

1.Functions; Domain and Range

??$$y=f(x)$$??????????????(“y equals f of x”)
??A function ? from a set D to a set Y is a rule that assigns a unique (single) element $f(x)\in Y$ to each element $x\in D$.
??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 ?(x).

2.Graphs of Functions

??If ? is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for ?. In set notation, the graph is$$\{(x,f(x))|x\in D\}$$

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 ? 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 ?, then the vertical line $x=a$ will intersect the graph of ? at the single point$(a,f(a))$ .

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 $?x?$ .
??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 $?x?$ .

6.Increasing and Decreasing Functions

7.Even Functions and Odd Functions: Symmetry

8.Common Functions

8.1 Linear Functions

??A function of the form $?(x)=mx+b$, for constants m and b, is
called a linear function.
??The function $?(x)=x$ where $m=1$ and $b=0$ 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 $y=kx$ for some nonzero constant k.

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

8.2 Power Functions

A function $f(x)=x^a$where a is a constant, is called a power function.

8.3 Polynomials

??A function p is a polynomial if$$p(x)=a_n{x}^n+a_{n?1}{x}^{n?1}+...+a_{1}x+a_0$$??where n is a nonnegative integer and the numbers $a_0,a_1,a_2...a_{n?1},a_n$are real constants(called the coefficients of the polynomial). If the leading coefficient an $a_n!=0$ and $n>0$, then n is called the degree of the polynomial.

8.4 Rational Functions

??A rational function is a quotient or ratio $?(x)=p(x)/q(x)$, where p and q are polynomials. The domain of a rational function is the set of all real x for which $q(x)!=0$

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 $f(x)=a^x$ , where the base $a>0$ is a positive constant and $a!=0$ have domain are called exponential functions.

8.8 Logarithmic Functions

??These are the functions $?(x)=lo{g}_a^x$, where the base $a!=1$ 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.

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