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New Words&Expressions:alphabet 字母表字母表 prime 素数，质数素数，质数displacement 位移位移 proportional 成比例的成比例的domain 定义域定义域 the real-valued function实值函数实值函数edge 棱，边棱，边 spring constant 弹性系数弹性系数 graph 图，图形图，图形 limit 极限极限 stretch 拉伸拉伸 volume 体积，容积，卷体积，容积，卷 2.6 函数的概念与函数思想函数的概念与函数思想Function concept and function ideaNew Words&Expressions(二二)P59:measurable 可测的可测的 mapping 映射映射 Various fields of human have to do with relationships that exist between one collection of objects and another.6-A Informal description of functions各行各业的人们都必须处理一类事物与另一类事物之各行各业的人们都必须处理一类事物与另一类事物之间存在的关系。间存在的关系。Graphs,charts,curves,tables,formulas,and Gallup polls are familiar to everyone who reads the newspapers.几几乎乎每每个个人人都都熟熟悉悉图图形形，图图表表，曲曲线线，公公式式和和盖盖洛洛普普民意测验。民意测验。These are merely devices for describing special relations in a quantitative fashion.Mathematicians refer to certain types of these relations as functions.这些只是以定量的方式描述特定关系的方法。数学家这些只是以定量的方式描述特定关系的方法。数学家将这些关系中的某些类型视作函数。将这些关系中的某些类型视作函数。In this section,we give an informal description of the function concept.A formal definition is given in Section 3.在在本本节节中中，我我们们给给出出一一个个非非正正式式的的描描述述函函数数的的概概念念。在第在第3节给出一个正式的定义节给出一个正式的定义。EXAMPLE 1.The force F necessary to stretch a steel spring a distance x beyond its natural length is proportional to x.That is,F=cx,where c is a number independent of x called the spring constant.把一个钢制的弹簧拉伸到超过其自然长度的距离为把一个钢制的弹簧拉伸到超过其自然长度的距离为x时所需要的力时所需要的力F与与x成正比。即，成正比。即，F=cx,这里这里c是不依赖是不依赖与与x的数，叫做弹性系数。的数，叫做弹性系数。This formula,discovered by Robert Hooke in the mid-17th century,is called Hookes law,and it is said to express the force as a function of the displacement.这个公式是在这个公式是在17世纪中叶被胡克发现的，叫做胡克定世纪中叶被胡克发现的，叫做胡克定律，它用来表示力关于位移的函数。律，它用来表示力关于位移的函数。EXAMPLE 2.The volume of a cube is a function of its edge-length.If the edges have length x,the volume V is given by the formula V=x3.立方体的体积是它棱长的函数。如果棱长为立方体的体积是它棱长的函数。如果棱长为x，那么，那么体积的公式为：体积的公式为：V=x3。EXAMPLE 3.A prime is any integer n1 that cannot be expressed in the form n=ab,where a and b are positive integers,both less than n.The first few primes are 2,3,5,7,11,13,17,19.素数是大于素数是大于1且不能表示成且不能表示成n=ab形式的整数，这里形式的整数，这里a和和b都是小于都是小于n的正整数。开始的几个素数是的正整数。开始的几个素数是2,3,5,7,11,13,17,19.For a given real number x0,it is possible to count the number of primes less than or equal to x.This number is said to be a function of x even though no simple algebraic formula is known for computing it(without counting)when x is known.对于一个给定的实数对于一个给定的实数x0，数出小于或者等于，数出小于或者等于x的素数的素数的个数是有可能的。这个数称为的个数是有可能的。这个数称为x的函数，尽管还没的函数，尽管还没有一个简单代数式可以由已知的有一个简单代数式可以由已知的x计算计算(不通过计数求不通过计数求)出它的值。出它的值。The word“function”was introduced into mathematics by Leibniz,who used the term primarily to refer to certain kinds of mathematical formulas.“函数函数”这个词是由莱布尼茨引入到数学中的，他主这个词是由莱布尼茨引入到数学中的，他主要使用这个术语来指代某种数学公式。要使用这个术语来指代某种数学公式。It was later realized that Leibnizs idea of function was much too limited in its scope,and the meaning of the word has since undergone many stages of generalization.后后来来人人们们才才认认识识到到，莱莱布布尼尼茨茨的的函函数数思思想想适适用用的的范范围围太太过过局局限限了了，这这个个术术语语的的含含义义从从那那时时起起已已经经过过了了多多次次推广。推广。Today,the meaning of function is essentially this:Given two sets,say X and Y,a function is a correspondence which associates with each element of X one and one only element of Y.如今，从本质上讲，函数的定义如下：给定两个集合如今，从本质上讲，函数的定义如下：给定两个集合X 和和Y，函数是，函数是X中元素与中元素与Y中元素的一一对应。中元素的一一对应。The set X is called the domain of the function.Those elements of Y associated with the elements in X form a set called the range of the function.(This may be all of Y,but it need not be)集集合合X叫叫做做函函数数的的定定义义域域，与与X中中的的元元素素相相对对应应的的Y中中的的元元素素的的集集合合叫叫做做函函数数的的值值域域。(值值域域可可能能是是整整个个集集合合Y,也可能不是。也可能不是。)Letters of the English and Greek alphabets are often used to denote functions.The particular letters f,g,h,F,G,H,and are frequently used for this purpose.英语字母和希腊字母表通常用于表示函数。为此，一英语字母和希腊字母表通常用于表示函数。为此，一些特定的字母如：些特定的字母如：f,g,h,频繁使用。频繁使用。If f is a given function and if x is an object of its domain,the notation f(x)is used to designate that object in the range which is associated to x by the function f;and it is called the value of f at x or the image of x under f.The symbol f(x)is read as“f of x.”如如果果f是是一一个个给给定定的的函函数数，x是是它它定定义义域域中中的的一一个个点点，符符号号f(x)表表示示值值域域中中按按照照f对对应应于于x的的点点，它它叫叫做做f在在x点点的值或者的值或者x在在f下的像。符号下的像。符号f(x)读作读作“f of x.”Seldom has a single concept played so important a role in mathematics as has the concept of function.It is desirable to know how the concept has developed.6-C The concept of function在数学中，很少有个概念象函数的概念那样，起那么在数学中，很少有个概念象函数的概念那样，起那么重要的作用的。因此，需要知道这个概念是如何发展重要的作用的。因此，需要知道这个概念是如何发展起来的。起来的。This concept,like many others,originates in physics.这个概念像许多其他概念一样，起源于物理学。这个概念像许多其他概念一样，起源于物理学。The physical quantities were the forerunners of mathematical variables,and relation among them was called a function relation in the later 16th century.物物理理量量是是数数学学变变量量的的先先驱驱，他他们们之之间间的的关关系系在在16世世纪纪后期称为函数关系。后期称为函数关系。For example,the formula s=16t2 for the number of feet s a body falls in any number of seconds t is a function relation between s and t,it describes the way s varies with t.例例如如,代代表表一一物物体体在在若若干干秒秒t中中下下落落若若干干英英尺尺s的的公公式式s=16t2 就就是是s和和t之之间间的的函函数数关关系系,它它描描述述了了s随随t 变变化化的的公式。公式。The study of such relations led people in the 18th century to think of a function relation as nothing but a formula.对对这这种种关关系系的的研研究究导导致致了了18世世纪纪的的人人们们认认为为函函数数关关系系只不过是一个公式罢了。只不过是一个公式罢了。Not specified by this definition is the manner of setting up the correspondence.至于如何建立这种对应关系，这个定义并未详细规定。至于如何建立这种对应关系，这个定义并未详细规定。It may be done by a formula as the 18th century mathematics presumed but it can equally well be done by a tabulation such as a statistical chart,or by some other form of description.可以如可以如18世纪的数学所假定的那样，用公式来建立，世纪的数学所假定的那样，用公式来建立，但同样也可以用统计表那样的表格或用某种其他的描但同样也可以用统计表那样的表格或用某种其他的描述方式来建立。述方式来建立。A typical example is the room temperature,which obviously is a function of time.But this function admits of no formula representation,although it can be recorded in a tabular form or traced out graphically by an automatic device.典型的例子是室温，这显然是一个时间的函数。但是典型的例子是室温，这显然是一个时间的函数。但是这个函数不能用公式来表示，不过可以用表格的形式这个函数不能用公式来表示，不过可以用表格的形式来记录或者用一种自动装置以图标形式来追踪。来记录或者用一种自动装置以图标形式来追踪。The modern definition of a function y of x is simply a mapping from a space X to another space Y.A mapping is defined when every point x of X has a definite image y,a point of Y.现现代代给给x的的一一个个函函数数y所所下下的的定定义义只只是是从从一一个个空空间间X到到另另一一个个空空间间Y的的映映射射。当当X空空间间的的每每一一个个点点x有有一一个个确确定的像点定的像点y，即，即Y空间的一点，那么映射就确定了。空间的一点，那么映射就确定了。The mapping concept is close to intuition,and therefore desirable to serve as a basis of the function concept.Moreover,as the space concept is incorporated in this modern definition,its generality contributes much to the generality of the function concept.这这个个映映射射概概念念接接近近于于直直观观，因因此此，值值得得作作为为函函数数概概念念的的一一个个基基础础。此此外外，由由于于这这个个现现代代的的定定义义中中体体现现了了空空间间的的概概念念，所所以以，它它的的普普及及性性对对函函数数概概念念的的普普及及性性有有很大的贡献。很大的贡献。谢 谢！
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