Point 1: properties and images of functions

函數(shù)的性質(zhì)是研究初等函數(shù)的基石，也是高考考查的重點(diǎn)內(nèi)容．在復(fù)習(xí)中要肯于在對(duì)定義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

復(fù)習(xí)函數(shù)的性質(zhì)，可以從“數(shù)”和“形”兩個(gè)方面，從理解函數(shù)的單調(diào)性和奇偶性的定義入手，在判斷和證明函數(shù)的性質(zhì)的問題中得以鞏固，在求復(fù)合函數(shù)的單調(diào)區(qū)間、函數(shù)的最值及應(yīng)用問題的過程中得以深化．具體要求是：

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．正確理解函數(shù)單調(diào)性和奇偶性的定義，能準(zhǔn)確判斷函數(shù)的奇偶性，以及函數(shù)在某一區(qū)間的單調(diào)性，能熟練運(yùn)用定義證明函數(shù)的單調(diào)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．從數(shù)形結(jié)合的角度認(rèn)識(shí)函數(shù)的單調(diào)性和奇偶性，深化對(duì)函數(shù)性質(zhì)幾何特征的理解和運(yùn)用，歸納總結(jié)求函數(shù)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．培養(yǎng)學(xué)生用運(yùn)動(dòng)變化的觀點(diǎn)分析問題，提高學(xué)生用換元、轉(zhuǎn)化、數(shù)形結(jié)合等數(shù)學(xué)思想方法解決問題的能力．

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

這部分內(nèi)容的重點(diǎn)是對(duì)函數(shù)單調(diào)性和奇偶性定義的深入理解．

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

函數(shù)的單調(diào)性只能在函數(shù)的定義域內(nèi)來(lái)討論．函數(shù)y＝f(x)在給定區(qū)間上的單調(diào)性，反映了函數(shù)在區(qū)間上函數(shù)值的變化趨勢(shì)，是函數(shù)在區(qū)間上的整體性質(zhì)，但不一定是函數(shù)在定義域上的整體性質(zhì)．函數(shù)的單調(diào)性是對(duì)某個(gè)區(qū)間而言的，所以要受到區(qū)間的限制．

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

對(duì)函數(shù)奇偶性定義的理解，不能只停留在f(－x)＝f(x)和f(－x)＝－f(x)這兩個(gè)等式上，要明確對(duì)定義域內(nèi)任意一個(gè)x，都有f(－x)＝f(x)，f(－x)＝－f(x)的實(shí)質(zhì)是：函數(shù)的定義域關(guān)于原點(diǎn)對(duì)稱．這是函數(shù)具備奇偶性的必要條件．稍加推廣，可得函數(shù)f(x)的圖象關(guān)于直線x＝a對(duì)稱的充要條件是對(duì)定義域內(nèi)的任意x，都有f(x＋a)＝f(a－x)成立．函數(shù)的奇偶性是其相應(yīng)圖象的特殊的對(duì)稱性的反映．

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

這部分的難點(diǎn)是函數(shù)的單調(diào)性和奇偶性的綜合運(yùn)用．根據(jù)已知條件，調(diào)動(dòng)相關(guān)知識(shí)，選擇恰當(dāng)?shù)姆椒ń鉀Q問題，是對(duì)學(xué)生能力的較高要求．

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

函數(shù)的圖象是函數(shù)性質(zhì)的直觀載體，函數(shù)的性質(zhì)可以通過函數(shù)的圖像直觀地表現(xiàn)出來(lái)。

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，掌握函數(shù)的圖像是學(xué)好函數(shù)性質(zhì)的關(guān)鍵，這也正是“數(shù)形結(jié)合思想”的體現(xiàn)。復(fù)習(xí)函數(shù)圖像要注意以下方面。

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．掌握描繪函數(shù)圖象的兩種基本方法——描點(diǎn)法和圖象變換法．

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

2．會(huì)利用函數(shù)圖象，進(jìn)一步研究函數(shù)的性質(zhì)，解決方程、不等式中的問題．

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

3．用數(shù)形結(jié)合的思想、分類討論的思想和轉(zhuǎn)化變換的思想分析解決數(shù)學(xué)問題．

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

4．掌握知識(shí)之間的聯(lián)系，進(jìn)一步培養(yǎng)觀察、分析、歸納、概括和綜合分析能力．

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

以解析式表示的函數(shù)作圖象的方法有兩種，即列表描點(diǎn)法和圖象變換法，掌握這兩種方法是本節(jié)的重點(diǎn)．

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

運(yùn)用描點(diǎn)法作圖象應(yīng)避免描點(diǎn)前的盲目性，也應(yīng)避免盲目地連點(diǎn)成線．要把表列在關(guān)鍵處，要把線連在恰當(dāng)處．這就要求對(duì)所要畫圖象的存在范圍、大致特征、變化趨勢(shì)等作一個(gè)大概的研究．而這個(gè)研究要借助于函數(shù)性質(zhì)、方程、不等式等理論和手段，是一個(gè)難點(diǎn)．用圖象變換法作函數(shù)圖象要確定以哪一種函數(shù)的圖象為基礎(chǔ)進(jìn)行變換，以及確定怎樣的變換．這也是個(gè)難點(diǎn)．

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