Point 1: properties and images of functions

函数的性质是研究初等函数的基石，也是高考考查的重点内容．在复习中要肯于在对定义的深入理解上下功夫．

The nature of function is the cornerstone of the study of elementary function and the key content of the college entrance examination. In the review, we should work hard on the in-depth understanding of the definition

复习函数的性质，可以从“数”和“形”两个方面，从理解函数的单调性和奇偶性的定义入手，在判断和证明函数的性质的问题中得以巩固，在求复合函数的单调区间、函数的最值及应用问题的过程中得以深化．具体要求是：

Reviewing the properties of functions can start with understanding the definitions of monotonicity and parity of functions from two aspects of "number" and "shape", consolidate in judging and proving the properties of functions, and deepen in the process of finding the monotonic interval of composite functions, the maximum value of functions and application problems. The specific requirements are as follows:

1．正确理解函数单调性和奇偶性的定义，能准确判断函数的奇偶性，以及函数在某一区间的单调性，能熟练运用定义证明函数的单调性和奇偶性．

1. Correctly understand the definition of monotonicity and parity of function, accurately judge the parity of function and the monotonicity of function in a certain interval, and skillfully use the definition to prove the monotonicity and parity of function

2．从数形结合的角度认识函数的单调性和奇偶性，深化对函数性质几何特征的理解和运用，归纳总结求函数最大值和最小值的常用方法．

2. Understand the monotonicity and parity of function from the perspective of the combination of number and shape, deepen the understanding and application of the geometric characteristics of function properties, and summarize the common methods of finding the maximum and minimum of function

3．培养学生用运动变化的观点分析问题，提高学生用换元、转化、数形结合等数学思想方法解决问题的能力．

3. Train students to analyze problems from the viewpoint of movement change, and improve students' ability to solve problems with mathematical ideas and methods such as yuan exchange, transformation and combination of numbers and shapes

这部分内容的重点是对函数单调性和奇偶性定义的深入理解．

The focus of this part is the in - depth understanding of the definitions of monotonicity and parity of functions

函数的单调性只能在函数的定义域内来讨论．函数y＝f(x)在给定区间上的单调性，反映了函数在区间上函数值的变化趋势，是函数在区间上的整体性质，但不一定是函数在定义域上的整体性质．函数的单调性是对某个区间而言的，所以要受到区间的限制．

The monotonicity of function can only be discussed in the domain of function definition. The monotonicity of function y = f (x) in a given interval reflects the change trend of function value in the interval. It is the overall property of function in the interval, but not necessarily the overall property of function in the domain of definition. The monotonicity of function is for a certain interval, so it is limited by the interval

对函数奇偶性定义的理解，不能只停留在f(－x)＝f(x)和f(－x)＝－f(x)这两个等式上，要明确对定义域内任意一个x，都有f(－x)＝f(x)，f(－x)＝－f(x)的实质是：函数的定义域关于原点对称．这是函数具备奇偶性的必要条件．稍加推广，可得函数f(x)的图象关于直线x＝a对称的充要条件是对定义域内的任意x，都有f(x＋a)＝f(a－x)成立．函数的奇偶性是其相应图象的特殊的对称性的反映．

To understand the definition of function parity, we should not only stay in the two equations of F (- x) = f (x) and f (- x) = f (x). It should be clear that for any X in the definition field, there is f (- x) = f (x), and the essence of F (- x) = f (x) is that the definition field of function is symmetrical about the origin. This is a necessary condition for the function to have parity. A little generalization, The necessary and sufficient condition for the symmetry of the image of function f (x) with respect to straight line x = a is that f (x + a) = f (A-X) holds for any X in the definition domain. The parity of function is the reflection of the special symmetry of its corresponding image

这部分的难点是函数的单调性和奇偶性的综合运用．根据已知条件，调动相关知识，选择恰当的方法解决问题，是对学生能力的较高要求．

The difficulty in this part is the comprehensive application of the monotonicity and parity of the function. According to the known conditions, mobilizing relevant knowledge and selecting appropriate methods to solve problems are high requirements for students' ability

函数的图象是函数性质的直观载体，函数的性质可以通过函数的图像直观地表现出来。

The image of function is the visual carrier of function properties, and the properties of function can be intuitively expressed through the image of function.

因此，掌握函数的图像是学好函数性质的关键，这也正是“数形结合思想”的体现。复习函数图像要注意以下方面。

Therefore, mastering the image of function is the key to learn the nature of function, which is the embodiment of "the combination of number and shape". When reviewing function images, pay attention to the following aspects.

1．掌握描绘函数图象的两种基本方法——描点法和图象变换法．

1. Master the two basic methods of drawing function images - point drawing method and image transformation method

2．会利用函数图象，进一步研究函数的性质，解决方程、不等式中的问题．

2. Be able to use function images to further study the properties of functions and solve problems in equations and inequalities

3．用数形结合的思想、分类讨论的思想和转化变换的思想分析解决数学问题．

3. Analyze and solve mathematical problems with the idea of combination of number and shape, classified discussion and transformation

4．掌握知识之间的联系，进一步培养观察、分析、归纳、概括和综合分析能力．

4. Master the connection between knowledge and further cultivate the ability of observation, analysis, induction, generalization and comprehensive analysis

以解析式表示的函数作图象的方法有两种，即列表描点法和图象变换法，掌握这两种方法是本节的重点．

There are two methods of making images with functions expressed analytically, namely, list point drawing method and image transformation method. Mastering these two methods is the focus of this section

运用描点法作图象应避免描点前的盲目性，也应避免盲目地连点成线．要把表列在关键处，要把线连在恰当处．这就要求对所要画图象的存在范围、大致特征、变化趋势等作一个大概的研究．而这个研究要借助于函数性质、方程、不等式等理论和手段，是一个难点．用图象变换法作函数图象要确定以哪一种函数的图象为基础进行变换，以及确定怎样的变换．这也是个难点．

When using the point drawing method to make an image, we should avoid the blindness before the point drawing, and we should also avoid blindly connecting points into lines. We should list the table at the key place and connect the lines at the appropriate place. This requires a general study on the existing range, general characteristics and change trend of the image to be drawn. This study should rely on the theories and means of function properties, equations, inequalities and so on, It is a difficult point. When using image transformation method to make function image, it is necessary to determine which kind of function image is the basis for transformation and what kind of transformation is determined. This is also a difficult point