专升本高数专题｜映射与函数
<h1 style="text-align: left; margin-bottom: 10px;">1. 映射</h1>
<h1 style="text-align: left; margin-bottom: 10px;">1.1 映射</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">映射：</span></strong> 集合</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">X</span></strong></span><span style="color: green;">有元素</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">x</span></strong></span><span style="color: green;">，如果存在一个法则</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">f</span></strong></span><span style="color: green;">，使得在</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">X</span></strong></span><span style="color: green;">中的任意元素</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">x</span></strong></span><span style="color: green;">都存在集合</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">Y</span></strong></span><span style="color: green;">中的元素</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">y</span></strong></span><span style="color: green;">与之一一对应。则称</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">f</span></strong></span><span style="color: green;">为从</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">X</span></strong></span><span style="color: green;">到</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">Y</span></strong></span><span style="color: green;">的<strong style="color: blue;"><span style="color: green;">映射</span></strong>。</span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/ef090546c27b4a1f8fb4822857d54936~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=aID%2BK5oWV%2B9i1qyk9DJLMCg7lB0%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;"> 像：</span></strong> 元素</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">y</span></strong></span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">原像：</span></strong> 元素</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">x</span></strong></span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">定义域：</span></strong> 集合</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">X</span></strong></span><span style="color: green;"><strong style="color: blue;"><span style="color: green;"> 值域：</span></strong> 集合</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">Y</span></strong></span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">满射</span></strong>就是</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">Y</span></strong></span><span style="color: green;">中所有元素都是</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">X</span></strong></span><span style="color: green;">中某元素的像，<strong style="color: blue;"><span style="color: green;">单射</span></strong>就是只存在一对一的映射，值得注意的是，映射又称为<strong style="color: blue;"><span style="color: green;">算子</span></strong>、<strong style="color: blue;"><span style="color: green;">泛函</span></strong>、<strong style="color: blue;"><span style="color: green;">变换</span></strong>，在不同的数学分支自然有不同的叫法，而在实数集到实数集的映射通常叫做<strong style="color: blue;"><span style="color: green;">函数</span></strong>。</span></p>
<h1 style="text-align: left; margin-bottom: 10px;">1.2 逆映射与复合映射</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">逆映射：</span></strong> </span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">f</span></strong></span><span style="color: green;">是从</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">X</span></strong></span><span style="color: green;">到</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">Y</span></strong></span><span style="color: green;">的<strong style="color: blue;"><span style="color: green;">单射</span></strong>，如果存在法则</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">g</span></strong></span><span style="color: green;">为从</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">Y</span></strong></span><span style="color: green;">到</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">X</span></strong></span><span style="color: green;">的映射，则法则</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">g</span></strong></span><span style="color: green;">是法则</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">f</span></strong></span><span style="color: green;">的逆映射。</span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">复合映射：</span></strong> </span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">f</span></strong></span><span style="color: green;">是从</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">X</span></strong></span><span style="color: green;">到</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">Y</span></strong></span><span style="color: green;">的<strong style="color: blue;"><span style="color: green;">映射</span></strong>，若</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">f</span></strong></span><span style="color: green;">的值域都在映射</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">g</span></strong></span><span style="color: green;">的定义域内，则</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">f</span></strong></span><span style="color: green;">与</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">g</span></strong></span><span style="color: green;">所构成的新映射为复合映射。</span></p>
<h1 style="text-align: left; margin-bottom: 10px;">2. 函数</h1>
<h1 style="text-align: left; margin-bottom: 10px;">2.1 函数的定义</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">函数：</span></strong> 定义域和值域都在<strong style="color: blue;"><span style="color: green;">实数域</span></strong>内的映射。</span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">自变量</span></strong>即映射的原像，<strong style="color: blue;"><span style="color: green;">因变量</span></strong>即映射的像。</span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">自然定义域：</span></strong> 对于用抽象的<strong style="color: blue;"><span style="color: green;">算式</span></strong>表达的函数，使得该算式有意义的一切实数所组成的集合。</span></p>
<h1 style="text-align: left; margin-bottom: 10px;">2.2 函数的几种特性</h1>
<h1 style="text-align: left; margin-bottom: 10px;">2.2.1 函数的有界性</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">上界：</span></strong> 对于一个函数</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">f(x)</span></strong></span><span style="color: green;">，存在实数</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">K1</span></strong></span><span style="color: green;">满足</span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/c4232eeb41f24563bfe68416708fb6a6~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=p3VR8RfWSff7mkjLyF95slfZQ7k%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">下界：</span></strong> 类比上界，</span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/1d6d042abbab4df29590b891fdd5ee37~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=q4DCgoXUUxB%2FUtOGxPMxoWD4C7U%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">有界：</span></strong> 存在正数</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">M</span></strong></span><span style="color: green;">，</span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/976545a944004e02b5b15c99e26377b0~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=aFy%2BU9WSSiiH%2F0bOrqWhLF3xFaQ%3D" style="width: 100%; margin-bottom: 20px;"></div>
<h1 style="text-align: left; margin-bottom: 10px;">2.2.2 函数的单调性</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">单调递增：</span></strong> 在一个区间内，变量值大的函数值总是大于变量值较小的函数值，即</span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/580782eae5b14f5494a031238314f949~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=yGNia8AzPBA%2BiOUxQ5JgjLz4bhg%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">单调递减：</span></strong> 在一个区间内，恒存在：</span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/3d89f779d8374f0ea4fd55f13ee3f630~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=96gkSjEzDPpqHOsJltS9A9DJ8SI%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">单调函数：</span></strong> 单调递增和单调递减的函数的统称。</span></p>
<h1 style="text-align: left; margin-bottom: 10px;">2.2.3 函数的奇偶性</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">奇函数：</span></strong></span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/2f6c3d2f47d04cfa863267e0f9cd2ac8~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=D0ckPnTB1BPdIvUcFXEnZskQ89Q%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">偶函数：</span></strong></span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/99db21e7bec2426a883f92fbc7c118e3~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=9Fkwhqtth2YFo%2FjJY4Vu%2FMkD4l8%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">非奇非偶：</span></strong> 同时不满足奇函数和偶函数的定义。</span></p>
<h1 style="text-align: left; margin-bottom: 10px;">2.2.4 函数的周期性</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">周期函数：</span></strong> 存在正数</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">T</span></strong></span><span style="color: green;">，使得函数</span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/07cfeeb97a624ba482167f6b362cd932~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=ZtGDJP7ZXWCo82df%2F1gCAFlcKL0%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">周期：</span></strong> 上述的正数</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">T</span></strong></span><span style="color: green;">，通常指<strong style="color: blue;"><span style="color: green;">最小正周期</span></strong>。</span></p>
<h1 style="text-align: left; margin-bottom: 10px;">2.3 反函数与复合函数</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">反函数：</span></strong> 定义域和值域都定义在实数集的映射</span><span style="color: green;"><strong style="color: blue;"><span style="color: green;">f</span></strong></span><span style="color: green;">的逆映射。</span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">复合函数：</span></strong> 定义域和值域都定义在实数集的映射所组成的复合映射。</span></p>
<h1 style="text-align: left; margin-bottom: 10px;">2.4 函数的运算</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;">和差商积。</span></p>
<h1 style="text-align: left; margin-bottom: 10px;">2.5 初等函数</h1>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">基本初等函数：</span></strong></span></p><span style="color: green;">指数函数</span><span style="color: green;">幂函数</span><span style="color: green;">对数函数</span><span style="color: green;">三角函数</span><span style="color: green;">反三角函数</span>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">初等函数：</span></strong> 用基本初等函数复合构成的函数。</span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/deca11b74b5b49c2b6b4986ae863b70b~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=BBnSWkZGw2ywxZkbTwE%2FU4RiqBE%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">知识点合集：</span></strong></span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">1. 形成映射的条件：</span></strong><span style="color: green;">A集合内的元素必须用完，对应关系为一对一或多对一。计算映射个数时，可以使用排列的思想来理解：A中第1个元素有几种对应情况×A中第2个元素有几种对应情况×…</span></span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">2. 原象和象：</span></strong><span style="color: green;">A→B，A中的元素为原象，B中的元素为象</span></span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">3. 函数解析式：</span></strong><span style="color: green;">对应关系，整体法带入的思想</span></span></p>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">1.1 映射与函数-题目</span></strong></span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/481c0e9494ac44c5b31760e7718e79e6~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=p%2BwxKWXYMxp9LhWW54RBqh1Z5oM%3D" style="width: 100%; margin-bottom: 20px;"></div>
<p style="font-size: 18px; line-height: 40px; text-align: left; margin-bottom: 30px;"><span style="color: green;"><strong style="color: blue;"><span style="color: green;">1.1 映射与函数-答案</span></strong></span></p>
<div style="text-align: left; margin-bottom: 10px;"><img src="https://p3-sign.toutiaoimg.com/tos-cn-i-qvj2lq49k0/626f0b82c4d142a7901ac7deda08812c~noop.image?_iz=58558&from=article.pc_detail&x-expires=1679805183&x-signature=NuidjDUNI9ZhG3W5g8e7fP6WL7k%3D" style="width: 100%; margin-bottom: 20px;"></div>
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