1.1）function，domain，range，the identity function，the absolute-value function，the real-valued function，real variable
2）cube，volume，edge-length，prime，totality
3）Hooke's law，stretch，displacement，spring，constant，proportional
4）schematic representation，plot，image，output，input
5）it is not difficult to imagine，the idea was much too limited
2.（1）常用英语字母和希腊字母来表示函数。
Letters of the English and Greek alphabets are often used to denote functions.
（2）若 f 是一个给定的函数，x 是定义域里的一个元素，那么记号f(x)用来表示由 f 确定的对应于x 的值。
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.
（3）该射线将两个坐标轴的夹角分成两个相等的角。
The ray makes equal angles with the coordinates axes.
（4）可以用许多方式给出函数思想的图解说明。
The function idea may be illustrated schematically in many ways.
（5）容易证明，绝对值函数满足三角不等式。
It is easy to proof that the absolute-value function satisfies the triangle inequality.
（6）对于实数x>0，函数g(x)表示不超过x 的素数的个数。
For a given real number x>0, the function g(x) is defined by the number of primes less than or equal to x.
（7）函数是一种对应，它未必可以表示成一个简单的代数公式。
A function is a correspondence. It is not necessary to be expressed by a simple algebraic formula. （8）在函数的定义中，关于定义域和值域中的对象，没对其性质做出任何限制。
The function idea places no restriction on the nature of the objects in the domain X and in the range Y.
3.
2.1 数学、方程与比例 （1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。 Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。 No modern scientific and technological branches could be regularly developed without the application of mathematics. （3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。 Before 17th century, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it. （7）方程很有用，可以用它来解决许多实际应用问题。 Equations are of very great use. We can use equations in many mathematical problems. （8）解方程时要进行一系列移项和同解变形，最后求出它的根，即未知量的值。To solve the equation means to move and change the terms about without making the equation untrue, until the root of the equation is obtained, which is the value of unknown term. 2.2 几何与三角 （1）许多专家都认为数学是学习其他科学技术的必备基础和先决条件。 Many experts recognize that mathematics is the necessary foundation and prerequisite of studying other science technology. （2）西方国家的专家认为几何起源于巴比伦和埃及人的土地测量技术，其实中国古代的数学家对几何做了许多出色的研究。 The western experts think that geometry had its origin in the measurements by the Babylonians and Egyptians of their lands. Infect, the ancient Chinese mathematicians made much remarkable study for geometry. （3）几何的学习使学生在思考问题时更周密和审慎，他们将不会盲目接受任何结论。 In studying geometry, the student is taught to think clearly and critically and he is led away from the practice of blind acceptance of any conclusions. （4）数学培养学生的分析问题的能力，使他们能应用毅力、创造性和逻辑推理来解决问题。
数学专业英语课后答案
2.1数学、方程与比例 词组翻译 1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation 2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction 3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart 4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation 5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction 6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place 汉译英 （1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。 Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。 No modern scientific and technological branches could be regularly developed without the application of mathematics. （3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。 Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。Before 17th century, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it. （7）方程很有用，可以用它来解决许多实际应用问题。
1) 史密斯太太对我抱怨说，她经常发现与自己十六岁的女儿简直无法沟通。 Mrs. Smith complained to me that she often found it simply impossible to communicate with her 16-year-old daughter. 2) 我坚信，阅读简写的 (simplified) 英文小说是扩大我们词汇量的一种轻松愉快的方法。 I firmly believe that reading simplified English novels is an easy and enjoyable way of enlarging our vocabulary. 3) 我认为我们在保护环境不受污染 (pollution) 方面还做得不够。 I don’t think we’re doing enough to protect our environment from pollution. 4) 除了每周写作文外，我们的英语老师还给我们布置了八本书在暑假里阅读。 In addition to/Apart from writing compositions on a weekly basis, our English teacher assigned us eight books to read during the summer vacation. 5) 我们从可靠的消息来源获悉下学期一位以英语为母语的人将要教我们英语口语。 We’ve learned from reliable sources that a native English speaker is going to teach us spoken English next term/semester. 6) 经常看英语电影不仅会提高你的听力，而且还会帮助你培养说的技能。 Seeing English movies on a regular basis will not only improve your ear, but will also help you build your speaking skills. 7) 如果你们对这些学习策略有什么问题，请随便问我。我将更详细地进行讲解。 If you have any questions about these learning strategies, please feel free to ask me. And I’ll explain them in greater detail. 8) 那个加拿大女孩善于抓住每个机会讲汉语。这就是她为什么三年不到就熟练地掌握了汉语口语的原因。 The Canadian girl is good at seizing every opportunity to speak Chinese. That’s why she has gained a good command of spoken Chinese in less than three years.
2.1数学、方程与比例 词组翻译 1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation 2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction 3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart 4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation 5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction 6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place 汉译英 （1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。 Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。 No modern scientific and technological branches could be regularly developed without the application of mathematics. （3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。 Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。 Before 17th century, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it. （7）方程很有用，可以用它来解决许多实际应用问题。
数学专业英语期中测试要求 将下列论文的摘要翻译为英语（按照学号顺序每个人分一个题目）。标题字体为黑体，字号三号，“摘要”二字以及“关键字”三字黑体，字号小四，其它字体为宋体，字号小四。行间距都为1.5倍行距。每人把自己要翻译的中文拷贝下来，然后在后面把它翻译为英文(英文字号与中文一样，字体为Times new roman，黑体的地方改为加粗)，写上姓名，学号（格式如下面的例子），打印出来交给我。大家认真做一下，你们写毕业论文要用到，学一学怎样写规范的文章，包括文章的排版（上学期反复强调交给我的matlab程序的打印的格式，结果有人交给我的程序竟然有上百页，字体很大，行距很大，还单面打印！（按照要求做的话只要几页就够了）有些人老是听不进老师讲的话！这样做既劳民伤财又不利于存档）。大家做好以后交到学习委员处，学习委员收齐以后交来给我，期末之前给我就行。 例子 中文标题 （姓名：学号：） 摘要 这是摘要正文…… …… 关键字：关键字1；关键字2；关键字3 English title （name: ,student id: ） Abstract This is the content of the abstract…… …… Keywords: keywords1; keywords2; keywords3;
1.泰勒公式的证明及其应用 摘要 本文先叙述并证明了不同条件下的泰勒公式，并进行了比较。然后讨论了泰勒公式在不同领域中的应用。本文着重论述了带有积分余项，拉格朗日余项和佩亚诺余项三种形式的泰勒公式，并比较了带不同余项的泰勒公式的异同。泰勒公式是数学分析中的重要知识，在理论分析和实际中都有广泛的应用。本文讨论了泰勒公式在10个方面的应用：利用泰勒公式求极限和导数，求无穷远处极限，证明中值公式，中值点的极限，证明不等式，导数的中值，估计关于界的估计，方程中的应用以及在近似计算中的应用。 关键词：泰勒公式；积分余项；佩亚诺余项；拉格朗日余项 2.论中学数学学习中的性别差异及其对成绩的影响 摘要 通过对初、高中两所学校男女学生数学成绩的调查，从智力、非智力、教育、社会环境等方面差异进行分析研究，得出性别差异对学生数学成绩有影响，并且随着年级的升高，男女生的数学成绩存在显著的差异。并提出了提高女生数学成绩应注意的几个方面。 关键词：中学生性别差异数学成绩因性施教 3.论在数学教学中培养学生的创新精神 摘要
大学英语精读课后翻译汇总 Unit 1 1)史密斯太太对我抱怨说，她经常发现与自己十六岁的女儿简直无法沟通。 Mrs. Smith complained to me that she often found it simply impossible to communicate with her 16-year-old daughter. 2) 我坚信，阅读简写的(simplified) 英文小说是扩大我们词汇量的一种轻松愉快的方法。 I firmly believe that reading simplified English novels is an easy and enjoyable way of enlarg ing our vocabulary. 3) 我认为我们在保护环境不受污染(pollution) 方面还做得不够。 I don’t think we’re doing enough to protect our environment from pollution. 4) 除了每周写作文外，我们的英语老师还给我们布置了八本书在暑假里阅读。 In addition to/Apart from writing compositions on a weekly basis, our English teacher assign ed us eight books to read during the summer vacation. 5) 我们从可靠的消息来源获悉下学期一位以英语为母语的人将要教我们英语口语。 We’ve learned from reliable sources that a native English speaker is going to teach us spok e n English next term/semester. 6) 经常看英语电影不仅会提高你的听力，而且还会帮助你培养说的技能。 Seeing English movies on a regular basis will not only improve your ear, but will also help y ou build your speaking skills. 7) 如果你们对这些学习策略有什么问题，请随便问我。我将更详细地进行讲解。 If you have any questions about these learning strategies, please feel free to ask me. And I’ll explain them in greater detail. 8) 那个加拿大女孩善于抓住每个机会讲汉语。这就是她为什么三年不到就熟练地掌握了汉语口语的原因。 The Canadian girl is good at seizing every opportunity to speak Chinese. That’s why she has gained a good command of spoken Chinese in less than three years. Unit 2 1) 幸好附近有家医院，我们立刻把他送到了那里。 Fortunately there was a hospital nearby and we took him there at once. 2) 胜利登上乔治岛(George Island) 后，船长向指挥部(the headquarters) 发了一份无线电报。After succeeding in landing on George Island, the captain sent a radio message to the head quarters. 3) 他决心继续他的实验，不过这一次他将用另一种方法来做。 He is determined to continue his experiment but this time he'll do it another way. 4) 她在读这部小说时，不禁想起了她在农村度过的那五年。 When she read the novel, she couldn't help thinking of the five years she had spent in the countryside. 5) 玛丽觉得单靠自己的力量执行她的计划是困难的。 Mary thought it difficult to carry out her plan all by herself. 6) 我们认为他不能在一刻钟内走完那段距离，但他却成功地做到了这一点。 We didn't think he could cover the distance in a quarter of an hour, but he succeeded in d oing it. 7) 甚至在他的医生告诉他患有肺癌之后，奇切斯特仍不肯放弃环球航行的宿愿。
“数学专业英语一次函数专题”教学设计 1. 教学内容及解析 教学内容：与直角坐标系及一次函数相关的英语文献资料 内容解析：本节内容应从两个方面去考虑. 一方面，从数学知识的角度，本节所涉及的是从平面直角坐标系到一次函数的数学知识，都是学生学过的基础知识及其应用，是对于这些知识的拉练式复习，并作为后面用英语叙述解答相关问题知识基础；另一方面，从语言的角度，学生会接触到大量没有接触过，但他们却很熟悉的英语词汇及短语，主动学习的学生可以在老师的指导下快速理解其含义并加入到自己的知识体系中，但需要花时间巩固，是需要读、写、说来巩固的内容. 这也正是本节的重点和难点所在. 2. 教学目标及解析 教学目标：①能根据已有的数学知识和给出的单词对照表，将给出的相关数学定理及结论的英语版本翻译为中文；②能理解并解答用英语表述的相关数学问题；③能将解决数学问题的过程用英语进行书面表述；④初步了解用英语表达与直角坐标系及一次函数相关数学理论的范式. 目标解析：学生掌握基本词汇，并能阅读与本节内容相关的英语文献，是学习和使用数学专业英语的最基本要求，学生需要在学习的过程中逐渐由接受，即阅读或聆听，逐渐向输出，即书面和口头表达的方向发展. 为达到此目标，学生需要在课前独立完成对于本节需要的单词的学习，并阅读一系列简短的相关数学文献，并在这个过程中初步体会数学专业英语的表达范式；在课堂上，在教师的指导下对自己的认知进一步补充，并进行一定量的练习，暴露自己的问题，在教师的帮助下修正问题，完善认知. 3. 教学问题诊断分析 学生在进行语言表达时，往往受到母语的限制，对于非母语的表达方式的接受能力一时难以习惯，而数学对于表达能力则有更高的要求，要求叙述简洁、逻辑清晰，因此本教学设计强调学生在有相当阅读量的积累后，通过自主练习，逐渐感受用英语表达数学理论的基本范式. 教师在教学过程中的主要任务在于为学生提供合适的学习资料，在学生学习的过程中给予提示与指导，及时指出学生的问题并予以纠正. 对于语言的学习，阅读量的积累是至关重要的，教师在课堂上的讲解、学生在课堂上的练习，都只是学习的一部分环节，如果需要帮助学生正确掌握表达数学理论的范式，尽量不留死角，就必须保证课前预习和课后复习的有效性，预习复习材料的充足性，这对教师的教学资料的积累和学生自主学习能力，都是一个挑战. 4. 教学支持条件分析 学情条件：初二8班是实验班，学生有较强的学习热情和好奇心，也知道课前预习、课上参与学习活动、课后复习的重要性，在曾经的数学课程教学中，也有一定的课前预习，课堂讨论的传统，对于本教学设计的各个环节，是可以理解教师的用意，并遵循教师的指导进行学习活动. 在知识层面上，学生已经系统学习过一次函数相关的知识，课程中提供的学习资料都是学生熟悉的数学知识；学生在英语学习上，已经有多年的积累，对于英语语法的学
Book2 Unit1 翻译 1.她砰地关上门，一声不吭地走了，他们之间那场争执就此结束。Their argument ended when she slammed the door and left without a word. 2. 出席晚宴的客人对那个美国人威严的语气感到有点意外。The guests at the dinner party were slightly surprised at the commanding tone of the American. 3. 约翰尼已长大成熟，不再害怕独自呆在家里了。 Johnny has outgrown the fear of staying at home alone. 4. 当全部乘客都向出口处(exit) 走去时，他却独自留在座位上，好像不愿意离开这架飞机似的。 While all the other passengers made for the exit, he alone remained in his seat as if unwilling to leave the plane. 5. 这封信必须交给威尔逊博士本人。 The letter is to be handed to Dr. Wilson himself. 6. 南希虽然很想参加辩论，但腼腆得不敢开口。 While she felt like joining in the argument, Nancy was too shy to open her mouth. 7. 你觉得什么时候最有可能在家里找到他？ What do you think is the likeliest time to find him at home?
一、数学符号翻译 12/156 twelve over one hundred and fifty-six 234/567 two hundred and thirty -four over five hundred and sixty-seven 4% four percent 7‰ seven permill 6.79 six point seven nine 52.3 three point two five recurring 85x the eighth root of x to the fifth power 53 x x to the minus three fifths dx x f b a )( f is integrable on [a,b] 32)(dx x f f is integrable on [2,3] x f the partial (derivative) of f with respect to x y f the partial (derivative) of f with respect to y 5lo g 3 the logarithm of five to the base three x a log the logarithm of x to the base a B A A cap B / A meet B / A intersection B B A A cup B / A join B / A union B A a a is an element of A B b b is an element of B 二、名词翻译 function theory 函数论 schematic representation 图解表示 geometry 几何学 trigonometry 三角学 consequence 推论 proposition 命题 theorem 定理 pyramid 棱锥 cone 圆锥 abscissa 横坐标 ordinate 纵坐标 coordinate system 坐标系 coordinate axis 坐标轴 polygonal region 多边形区域 tangent 切线 slope 斜率 complex-valued sequence 复值序列 convergence 收敛 real valued sequence 实值序列 divergence 发散 rational number 有理数 irrational number 无理数 odd number 奇数 even number 偶数 prime 质数、素数 composite 合数 plane 平面 surface 曲面 broken line 折线 Venn diagram 文氏图 universal set 全集 the underlying set 基础集 proper subset 真子集 quadratic equation 二次方程 the absolute-value function 绝对值函数
大学英语精读第三版第四册课后翻译答案 Unit1翻译 1) 我们接到通知，财政部长将于次日接见我们。 We were informed that the Minister of Finance was to give us an audience /receive us the next day. 2) 我觉得很奇怪，他似乎不记得自己的生日。 I thought it odd that he didn't seem to remember his own birthday. 3) 学期论文最迟应在下星期二交来，可是至今大部分学生却几无进展。 Next Tuesday is the deadline for handing in the term papers, but most students have hardly made a dent in the work so far. 4) 看到学生人数不断减少，校长心里很难受。(pain) It pained the headmaster to find the number of students shrinking. 5) 在那个国家一般用现金付账，但支票变得普遍起来了，不久会代替现金作为人们结账的 一种方式。 Cash is commonly used in paying bills in that country, but checks are becoming more popular and will, in a short while, replace cash as a way for people to settle their accounts. 6) 该公司声称，这条河流的污染不是它造成的。 The company claims that it is not responsible for the pollution in the river. Unit2翻译 1) 比尔已是个成熟的小伙子，不再依赖父母替他做主。 Bill is a mature young man who is no longer dependent on his parents for decisions. 2) 这个地区有大量肉类供应，但新鲜果蔬奇缺。 There are abundant supplies of meat in this region, but fresh fruit and vegetables are scarce. 3) 工程师们依靠工人们的智慧发明了一种新的生产方法，使生产率得以提高。
第一章 数学专业英语的阅读和翻译初级阶段 1.1 数学专业英语的基本特点 一 注意对客观事实与真理的描述： 1. 语句事态的使用上常用一般现在时 例 An equation is a statement of the equality between two equal numbers or number symbols. 2. 被动语态出现频率高“It is …”句型也使用得多 例 It is cleat that any function defined for all positive real x may be used to construct a sequence by restricting x to take only integer values. 3. 主动语态句型也多数用于强调事实，而不是强调行为发出者及其情感 例 1：Given ε> 0，there exits a number N>0. such that ε<-a a n for all n ≥N 例 2：Since h(x) is harmonic on a neighborhood of B(a,r), we have )()()(a h x d x h B =? ?σ “we have ” =one has 可省略 译为 “可以得出”什么结论 二 科学内容的完整性与表达形式的精炼性要求 三 数学的专业性十分典型 1、有的概念可能有几个同义词 如“计算”有： count calculate calculation calculus compute computation 2、同一词根的次和词组很多，如： Integrability 可积性 integrable 可积的 integral 积分，积分的，整数的 integral calculus 积分学 integralization 整化 integrate 积分 3、半专业词汇多是出现频率高 如: function 函数 functional 泛函 power 幂 set 假定，令
1－A What is mathematics Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. And in turn, mathematics serves the practice and plays a great role in all fields. No modern scientific and technological branches could be regularly developed without the application of mathematics. 数学来源于人类的社会实践，比如工农业生产，商业活动，军事行动和科学技术研究。反过来，数学服务于实践，并在各个领域中起着非常重要的作用。没有应用数学，任何一个现在的科技的分支都不能正常发展。From the early need of man came the concepts of numbers and forms. Then, geometry developed out of problems of measuring land , and trigonometry came from problems of surveying . To deal with some more complex practical problems, man established and then solved equation with unknown numbers ,thus algebra occurred. Before 17th century, man confined himself to the elementary mathematics, . , geometry, trigonometry and algebra, in which only the constants are considered. 很早的时候，人类的需要产生了数和形式的概念，接着，测量土地的需要形成了几何，出于测量的需要产生了三角几何，为了处理更复杂的实际问题，人类建立和解决了带未知参数的方程，从而产生了代数学，17世纪前，人类局限于只考虑常数的初等数学，即几何，三角几何和代数。The rapid development of industry in 17th century promoted the progress of economics and technology and required dealing with variable quantities. The leap from constants to variable quantities brought about two new branches of mathematics----analytic geometry and calculus, which belong to the higher mathematics. Now there are many branches in higher mathematics, among which are mathematical analysis, higher algebra, differential equations, function theory and so on. 17世纪工业的快速发展推动了经济技术的进步，从而遇到需要处理变量的问题，从常数带变量的跳跃产生了两个新的数学分支-----解析几何和微积分，他们都属于高等数学，现在高等数学里面有很多分支，其中有数学分析，高等代数，微分方程，函数论等。Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful
1.1）function，domain，range，the identity function，the absolute-value function，the real-valued function，real variable 2）cube，volume，edge-length，prime，totality 3）Hooke's law，stretch，displacement，spring，constant，proportional 4）schematic representation，plot，image，output，input 5）it is not difficult to imagine，the idea was much too limited 2.（1）常用英语字母和希腊字母来表示函数。 Letters of the English and Greek alphabets are often used to denote functions. （2）若 f 是一个给定的函数，x 是定义域里的一个元素，那么记号f(x)用来表示由 f 确定的对应于x 的值。 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. （3）该射线将两个坐标轴的夹角分成两个相等的角。 The ray makes equal angles with the coordinates axes. （4）可以用许多方式给出函数思想的图解说明。 The function idea may be illustrated schematically in many ways. （5）容易证明，绝对值函数满足三角不等式。 It is easy to proof that the absolute-value function satisfies the triangle inequality. （6）对于实数x>0，函数g(x)表示不超过x 的素数的个数。 For a given real number x>0, the function g(x) is defined by the number of primes less than or equal to x. （7）函数是一种对应，它未必可以表示成一个简单的代数公式。 A function is a correspondence. It is not necessary to be expressed by a simple algebraic formula. （8）在函数的定义中，关于定义域和值域中的对象，没对其性质做出任何限制。 The function idea places no restriction on the nature of the objects in the domain X and in the range Y. 3.