From fe04675d9980ef6d459ca28aded3f2b4756cc42e Mon Sep 17 00:00:00 2001 From: cyrxdzj <2824671438@qq.com> Date: Mon, 9 Oct 2023 21:10:27 +0800 Subject: [PATCH] Site updated: 2023-10-09 21:10:27 --- .../index.html | 2 +- 2021-07-07-HowToUseLuoguCard/index.html | 2 +- .../index.html | 2 +- .../index.html | 2 +- .../index.html | 2 +- .../index.html | 2 +- 2023-09-22-CyrxNote-Math-001/index.html | 2 +- 2023-09-23-CyrxNote-Math-002/index.html | 2 +- 2023-09-28-CyrxNote-Chemistry-001/index.html | 2 +- 2023-10-04-CyrxNote-Math-003/index.html | 2 +- 2023-10-09-CyrxNote-Math-004/index.html | 32 ++++++++++++++++-- 2023-10-09-CyrxNoteIndex/index.html | 4 +-- 404/index.html | 2 +- about/index.html | 2 +- archives/2021/07/index.html | 2 +- archives/2021/index.html | 2 +- archives/2023/05/index.html | 2 +- archives/2023/09/index.html | 2 +- archives/2023/10/index.html | 2 +- archives/2023/index.html | 2 +- archives/index.html | 2 +- atom.xml | 8 ++--- baidusitemap.xml | 4 +-- .../index.html" | 2 +- categories/index.html | 2 +- .../index.html" | 2 +- .../index.html" | 2 +- .../\346\235\202\350\260\210/index.html" | 2 +- .../index.html" | 2 +- friends/index.html | 2 +- index.html | 2 +- .../CyrxNote-Math-004-02.PNG | Bin 0 -> 29307 bytes .../CyrxNote-Math-004-03.PNG | Bin 0 -> 43470 bytes search.xml | 4 +-- sitemap.txt | 4 +-- sitemap.xml | 4 +-- tags/index.html | 2 +- 37 files changed, 72 insertions(+), 44 deletions(-) create mode 100644 post_image/CyrxNote-Math-004/CyrxNote-Math-004-02.PNG create mode 100644 post_image/CyrxNote-Math-004/CyrxNote-Math-004-03.PNG diff --git a/2021-07-02-LuoguMonthlyCompetition202105Summary/index.html b/2021-07-02-LuoguMonthlyCompetition202105Summary/index.html index a0eec38c..372a8263 100644 --- a/2021-07-02-LuoguMonthlyCompetition202105Summary/index.html +++ b/2021-07-02-LuoguMonthlyCompetition202105Summary/index.html @@ -1065,7 +1065,7 @@

 站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2021-07-07-HowToUseLuoguCard/index.html b/2021-07-07-HowToUseLuoguCard/index.html index 74d98488..26d779e8 100644 --- a/2021-07-07-HowToUseLuoguCard/index.html +++ b/2021-07-07-HowToUseLuoguCard/index.html @@ -923,7 +923,7 @@

 站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2021-07-09-Grade7Term2FinalExamSummary/index.html b/2021-07-09-Grade7Term2FinalExamSummary/index.html index dfd338be..9dedab00 100644 --- a/2021-07-09-Grade7Term2FinalExamSummary/index.html +++ b/2021-07-09-Grade7Term2FinalExamSummary/index.html @@ -882,7 +882,7 @@

 站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2021-07-09-Grade7Term2MidtermExamSummary/index.html b/2021-07-09-Grade7Term2MidtermExamSummary/index.html index 4474659a..2f999214 100644 --- a/2021-07-09-Grade7Term2MidtermExamSummary/index.html +++ b/2021-07-09-Grade7Term2MidtermExamSummary/index.html @@ -867,7 +867,7 @@

 站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2021-07-22-LuoguMonthlyCompetition202107Summary/index.html b/2021-07-22-LuoguMonthlyCompetition202107Summary/index.html index b6bfd9d3..8d7df53b 100644 --- a/2021-07-22-LuoguMonthlyCompetition202107Summary/index.html +++ b/2021-07-22-LuoguMonthlyCompetition202107Summary/index.html @@ -1189,7 +1189,7 @@

 站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2023-05-21-TalkingAboutComputerSecurityFromTheIntelligentClassromSystem/index.html b/2023-05-21-TalkingAboutComputerSecurityFromTheIntelligentClassromSystem/index.html index 7f237575..312b682c 100644 --- a/2023-05-21-TalkingAboutComputerSecurityFromTheIntelligentClassromSystem/index.html +++ b/2023-05-21-TalkingAboutComputerSecurityFromTheIntelligentClassromSystem/index.html @@ -858,7 +858,7 @@

 站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2023-09-22-CyrxNote-Math-001/index.html b/2023-09-22-CyrxNote-Math-001/index.html index 8dc1811f..6dd8d3c6 100644 --- a/2023-09-22-CyrxNote-Math-001/index.html +++ b/2023-09-22-CyrxNote-Math-001/index.html @@ -919,7 +919,7 @@

 站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2023-09-23-CyrxNote-Math-002/index.html b/2023-09-23-CyrxNote-Math-002/index.html index 6e7620b9..f04c50fc 100644 --- a/2023-09-23-CyrxNote-Math-002/index.html +++ b/2023-09-23-CyrxNote-Math-002/index.html @@ -915,7 +915,7 @@

  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2023-09-28-CyrxNote-Chemistry-001/index.html b/2023-09-28-CyrxNote-Chemistry-001/index.html index d0c5bd34..f6dd08cb 100644 --- a/2023-09-28-CyrxNote-Chemistry-001/index.html +++ b/2023-09-28-CyrxNote-Chemistry-001/index.html @@ -1008,7 +1008,7 @@

花絮   站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2023-10-04-CyrxNote-Math-003/index.html b/2023-10-04-CyrxNote-Math-003/index.html index 4ab13241..cbc742fd 100644 --- a/2023-10-04-CyrxNote-Math-003/index.html +++ b/2023-10-04-CyrxNote-Math-003/index.html @@ -1035,7 +1035,7 @@

 站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2023-10-09-CyrxNote-Math-004/index.html b/2023-10-09-CyrxNote-Math-004/index.html index de22ef71..1836e0fb 100644 --- a/2023-10-09-CyrxNote-Math-004/index.html +++ b/2023-10-09-CyrxNote-Math-004/index.html @@ -454,7 +454,7 @@

cyrxdzj的文化课学习笔记
文章字数:   - 719 + 1.4k
@@ -498,6 +498,34 @@

可以计算得出，当 $x=0$ 时，函数值为 $0$。当 $x=2$ 时，函数值取到最小值，为 $-4$。当 $x=4$ 时，函数值再次为 $0$。

因此，可以得出 $2\le m\le4$。如果 $m$ 太小，函数值就不够小；如果 $m$ 太大，函数值就会超过 $0$。

+

3、当给出的函数不标准的时候

有些问题给出的函数，不会给 $f(x)$ 这种标准的形式，而是各种各样乱七八糟的形式。

+

已知 $f(2x-1)=4x^2$，则下列结论中正确的是：

+

A. $f(3)=-9$

+

B. $f(-3)=-4$

+

C. $f(x)=x^2$

+

D. $f(x)=(x+1)^2$

+

对于这类问题，首先为了避免混淆，先将题干中给出的函数变量改个名字，变成 $f(2a-1)=4a^2$。

+

然后，对于每个选项，挨个儿解出函数括号内的数对应的 $a$ 是多少。

+

例如，A 选项，我们可得 $2a-1=3$，解得 $a=2,4a^2=16\neq-9$。A 选项错误。

+

B 选项，我们可得 $2a-1=-3$，解得 $a=-1,4a^2=4\neq-4$，B 选项错误。

+

C 和 D 选项，我们可得 $2a-1=x$，稍稍细心计算，得 $a=\frac{x+1}{2}$，由此可得 $4a^2=(x+1)^2$，D 选项正确而 C 选项错误。

+

对于此类问题，应将题干中给出的原函数传入的参数（即 $2a-1$）和选项中给出的待验证函数传入的参数（即 $3$、$-3$、$x$）进行比对，要求两者相等，得出原函数传入的参数中的字母具体是什么（即，当 $2a-1=3$ 时 $a=2$），然后将字母代入计算。为避免混淆，可以将原函数中的字母统一更名。

+

4、仔细分析更复杂的函数

设 $f(x)=\frac{1+x^2}{1-x^2}$，则下列结论正确的是？请注意本题多选。

+

A. $f(-x)=-f(x)$

+

B. $f(\frac{1}{x})=-f(x)$

+

C. $f(-\frac{1}{x})=f(x)$

+

D. $f(-x)=f(x)$

+

我们可以发现，$x^2=(-x)^2$，$x$ 的正负性并不影响平方的结果。因此，$\frac{1+x^2}{1-x^2}=\frac{1+(-x)^2}{1-(-x)^2}$，由此可得 $f(-x)=f(x)$，A 选项错误，D 选项正确。

+

顺便说一句，这样的函数叫做偶函数，函数关于 $y$ 轴对称，图象如下：

+

+

同时，我们可以得到 $f(\frac{1}{x})=f(-\frac{1}{x})$，这一条待会有用。

+

接下来，细心推出以下式子：

+

$f(\frac{1}{x})\\=\frac{1+\frac{1}{x}^2}{1-\frac{1}{x}^2}\\=\frac{x^2+1}{x^2}\div\frac{x^2-1}{x^2}\\=\frac{1+x^2}{x^2-1}$

+

至此，函数 $f(\frac{1}{x})$ 计算完毕。

+

而函数 $f(x)=\frac{1+x^2}{1-x^2}$，可以很容易地发现 $f(\frac{1}{x})=-f(x)$，B 选项正确，C 选项错误。图象如下：

+

+

可以恰好发现 $f(x)$ 与 $f(\frac{1}{x})$ 关于 $x$ 轴对称。

+

对于此类问题，应细心审题，谨慎作答，答后验证。

@@ -870,7 +898,7 @@

 站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/2023-10-09-CyrxNoteIndex/index.html b/2023-10-09-CyrxNoteIndex/index.html index 92d5be37..137b0bd3 100644 --- a/2023-10-09-CyrxNoteIndex/index.html +++ b/2023-10-09-CyrxNoteIndex/index.html @@ -508,7 +508,7 @@

cyrxdzj的文化课学习笔记 数学 cyrxdzj的文化课学习笔记 数学004 函数 2023-10-09 07:01:12 -2023-10-09 18:48:51 +2023-10-09 21:08:29 化学 @@ -884,7 +884,7 @@

cyrxdzj的文化课学习笔记
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/404/index.html b/404/index.html index 027353ae..4b14d201 100644 --- a/404/index.html +++ b/404/index.html @@ -344,7 +344,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/about/index.html b/about/index.html index a95ee8c9..8571d000 100644 --- a/about/index.html +++ b/about/index.html @@ -801,7 +801,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/archives/2021/07/index.html b/archives/2021/07/index.html index 68f390c3..4478ab3b 100644 --- a/archives/2021/07/index.html +++ b/archives/2021/07/index.html @@ -738,7 +738,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/archives/2021/index.html b/archives/2021/index.html index c21323c6..ea9a4683 100644 --- a/archives/2021/index.html +++ b/archives/2021/index.html @@ -738,7 +738,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/archives/2023/05/index.html b/archives/2023/05/index.html index f7840886..177393ed 100644 --- a/archives/2023/05/index.html +++ b/archives/2023/05/index.html @@ -518,7 +518,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/archives/2023/09/index.html b/archives/2023/09/index.html index 6d9e1005..7281aaee 100644 --- a/archives/2023/09/index.html +++ b/archives/2023/09/index.html @@ -632,7 +632,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/archives/2023/10/index.html b/archives/2023/10/index.html index 1ddacca7..bae80ad1 100644 --- a/archives/2023/10/index.html +++ b/archives/2023/10/index.html @@ -642,7 +642,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/archives/2023/index.html b/archives/2023/index.html index f8e15cf4..7b427d0a 100644 --- a/archives/2023/index.html +++ b/archives/2023/index.html @@ -876,7 +876,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/archives/index.html b/archives/index.html index 0471aac6..79086e84 100644 --- a/archives/index.html +++ b/archives/index.html @@ -1161,7 +1161,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/atom.xml b/atom.xml index f918cfbf..120df38a 100644 --- a/atom.xml +++ b/atom.xml @@ -6,7 +6,7 @@ - 2023-10-09T10:50:51.009Z + 2023-10-09T13:08:54.594Z https://blog.cyrxdzj.eu.org/ @@ -20,8 +20,8 @@ cyrxdzj的文化课学习笔记 目录 https://blog.cyrxdzj.eu.org/2023-10-09-CyrxNoteIndex/ - 2023-10-09T10:50:51.000Z - 2023-10-09T10:50:51.009Z + 2023-10-09T13:08:54.000Z + 2023-10-09T13:08:54.594Z @@ -45,7 +45,7 @@ https://blog.cyrxdzj.eu.org/2023-10-09-CyrxNote-Math-004/ 2023-10-08T23:01:12.000Z - 2023-10-09T10:50:45.993Z + 2023-10-09T13:09:35.132Z diff --git a/baidusitemap.xml b/baidusitemap.xml index 95747889..b27dde7a 100644 --- a/baidusitemap.xml +++ b/baidusitemap.xml @@ -1,7 +1,7 @@ - https://blog.cyrxdzj.eu.org/2023-10-04-CyrxNote-Math-003/ + https://blog.cyrxdzj.eu.org/2023-10-09-CyrxNote-Math-004/ 2023-10-09 @@ -9,7 +9,7 @@ 2023-10-09 - https://blog.cyrxdzj.eu.org/2023-10-09-CyrxNote-Math-004/ + https://blog.cyrxdzj.eu.org/2023-10-04-CyrxNote-Math-003/ 2023-10-09 diff --git "a/categories/cyrxdzj\347\232\204\346\226\207\345\214\226\350\257\276\345\255\246\344\271\240\347\254\224\350\256\260/index.html" "b/categories/cyrxdzj\347\232\204\346\226\207\345\214\226\350\257\276\345\255\246\344\271\240\347\254\224\350\256\260/index.html" index 6f96e776..ed9c8d61 100644 --- "a/categories/cyrxdzj\347\232\204\346\226\207\345\214\226\350\257\276\345\255\246\344\271\240\347\254\224\350\256\260/index.html" +++ "b/categories/cyrxdzj\347\232\204\346\226\207\345\214\226\350\257\276\345\255\246\344\271\240\347\254\224\350\256\260/index.html" @@ -689,7 +689,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/categories/index.html b/categories/index.html index bba8852e..cd97a843 100644 --- a/categories/index.html +++ b/categories/index.html @@ -481,7 +481,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git "a/categories/\344\270\200\344\272\233\346\235\202\344\270\203\346\235\202\345\205\253\347\232\204\344\270\234\350\245\277/index.html" "b/categories/\344\270\200\344\272\233\346\235\202\344\270\203\346\235\202\345\205\253\347\232\204\344\270\234\350\245\277/index.html" index def5d275..4c4e1749 100644 --- "a/categories/\344\270\200\344\272\233\346\235\202\344\270\203\346\235\202\345\205\253\347\232\204\344\270\234\350\245\277/index.html" +++ "b/categories/\344\270\200\344\272\233\346\235\202\344\270\203\346\235\202\345\205\253\347\232\204\344\270\234\350\245\277/index.html" @@ -471,7 +471,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git "a/categories/\345\255\246\346\240\241\347\224\237\346\264\273/index.html" "b/categories/\345\255\246\346\240\241\347\224\237\346\264\273/index.html" index 7e3c1568..52dadb6b 100644 --- "a/categories/\345\255\246\346\240\241\347\224\237\346\264\273/index.html" +++ "b/categories/\345\255\246\346\240\241\347\224\237\346\264\273/index.html" @@ -511,7 +511,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git "a/categories/\346\235\202\350\260\210/index.html" "b/categories/\346\235\202\350\260\210/index.html" index ce03e78a..c075d0aa 100644 --- "a/categories/\346\235\202\350\260\210/index.html" +++ "b/categories/\346\235\202\350\260\210/index.html" @@ -471,7 +471,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git "a/categories/\346\257\224\350\265\233\346\200\273\347\273\223/index.html" "b/categories/\346\257\224\350\265\233\346\200\273\347\273\223/index.html" index 19e3e1b4..63527612 100644 --- "a/categories/\346\257\224\350\265\233\346\200\273\347\273\223/index.html" +++ "b/categories/\346\257\224\350\265\233\346\200\273\347\273\223/index.html" @@ -511,7 +511,7 @@
  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/friends/index.html b/friends/index.html index 5a5f2a8e..9cdba906 100644 --- a/friends/index.html +++ b/friends/index.html @@ -650,7 +650,7 @@

dengziyue


  站点总字数: 14k 字 + class="white-color">14.6k 字 diff --git a/index.html b/index.html index f2cff80b..ece618e9 100644 --- a/index.html +++ b/index.html @@ -1160,7 +1160,7 @@

如何使用luogu-card


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-8,7 +8,7 @@ /2023-10-09-CyrxNoteIndex/ - 此为《cyrxdzj的文化课学习笔记》的目录。您可以在这里寻找您需要的文章。

科目标题创建时间更新时间
数学cyrxdzj的文化课学习笔记 数学001 基本不等式2023-09-22 21:58:372023-10-05 07:36:34
数学cyrxdzj的文化课学习笔记 数学002 集合与条件2023-09-23 22:05:132023-10-03 19:27:18
数学cyrxdzj的文化课学习笔记 数学003 一元二次不等式2023-10-04 20:57:512023-10-08 18:45:00
数学cyrxdzj的文化课学习笔记 数学004 函数2023-10-09 07:01:122023-10-09 18:48:51
化学cyrxdzj的文化课学习笔记 化学001 氧化还原反应2023-09-28 20:25:462023-09-28 21:08:06
]]> + 此为《cyrxdzj的文化课学习笔记》的目录。您可以在这里寻找您需要的文章。

科目标题创建时间更新时间
数学cyrxdzj的文化课学习笔记 数学001 基本不等式2023-09-22 21:58:372023-10-05 07:36:34
数学cyrxdzj的文化课学习笔记 数学002 集合与条件2023-09-23 22:05:132023-10-03 19:27:18
数学cyrxdzj的文化课学习笔记 数学003 一元二次不等式2023-10-04 20:57:512023-10-08 18:45:00
数学cyrxdzj的文化课学习笔记 数学004 函数2023-10-09 07:01:122023-10-09 21:08:29
化学cyrxdzj的文化课学习笔记 化学001 氧化还原反应2023-09-28 20:25:462023-09-28 21:08:06
]]> @@ -27,7 +27,7 @@ /2023-10-09-CyrxNote-Math-004/ - 1、思维杂题

已知 $A=B=${$0,1,2,3$}，则 $f:A\rightarrow B$ 有多少种值域？

可以发现，值域一定是集合 $B$ 的子集，同时值域不能为空集。而一个集合的非空子集的数量是 $2^n-1$ 个，所以值域有 $15$ 种。

若两个函数的对应关系相同，值域也相同，但定义域不同，则称这两个函数为“同族函数”。那么，函数 $y=x^2,x\in${$-1,0,1,2$} 有多少个同族函数？

可以很容易地得到，该函数的值域为 {$0,1,4$}，由此可知定义域符合如下性质：

  1. $0$ 必须出现；
  2. $\pm1$ 至少出现一个，也可以都出现，共 $3$ 种可能；
  3. $\pm2$ 同上；
  4. 其它数均不能出现。

综上，相同的值域，定义域有 $9$ 种可能。

但是题中要求定义域不同，而上面 $9$ 种定义域有一种其实就是题中提到的。因此，答案为 $8$。

已知函数 $f(x)$ 的定义域为 {$1,2,3,4$}，值域为 {$7,8,9$}，且对任意的 $x<y$ 都有 $f(x)\le f(y)$，则满足条件的不同函数共有多少个？

值域中每一个数，都至少有一个定义域中的数与之对应；而定义域中的每一个数，都恰好对应一个值域中的数。分析题目，题中要求定义域中有 $4$ 个数，可值域只有 $3$ 个数，换句话说，恰好有 $2$ 个定义域的数，对应的值域中的数，是同一个。

而题中还说 $x<y,f(x)\le f(y)$，结合上文，我们可以列出不等式：

这个式子中的 $3$ 个不等号有且只有 $1$ 个实际上是取等号的。如果有大于 $1$ 个不等号实际上取等号，就会导致函数可能的值的数量小于 $3$，值域中有数找不到定义域中的数与之对应；而如果没有不等号实际上取等号，就会导致 $4$ 个 $x$ 产生了 $4$ 个 $f(x)$，可值域的大小才 $3$ 个。

而任何一个不等号实际上取了等号，我们都能构造出对应关系。

因此，满足条件的不同函数，有 $3$ 个。

2、与一元二次函数结合起来

前置知识

已知函数 $f(x)=x^2-4x$ 在 $[0,m]$ 上的值域为 $[-4,0]$，则实数 $m$ 的取值范围是？

首先，通过图象，我们可以看出 $x^2-4x$ 的情况：

可以计算得出，当 $x=0$ 时，函数值为 $0$。当 $x=2$ 时，函数值取到最小值，为 $-4$。当 $x=4$ 时，函数值再次为 $0$。

因此，可以得出 $2\le m\le4$。如果 $m$ 太小，函数值就不够小；如果 $m$ 太大，函数值就会超过 $0$。

]]> + 1、思维杂题

已知 $A=B=${$0,1,2,3$}，则 $f:A\rightarrow B$ 有多少种值域？

可以发现，值域一定是集合 $B$ 的子集，同时值域不能为空集。而一个集合的非空子集的数量是 $2^n-1$ 个，所以值域有 $15$ 种。

若两个函数的对应关系相同，值域也相同，但定义域不同，则称这两个函数为“同族函数”。那么，函数 $y=x^2,x\in${$-1,0,1,2$} 有多少个同族函数？

可以很容易地得到，该函数的值域为 {$0,1,4$}，由此可知定义域符合如下性质：

  1. $0$ 必须出现；
  2. $\pm1$ 至少出现一个，也可以都出现，共 $3$ 种可能；
  3. $\pm2$ 同上；
  4. 其它数均不能出现。

综上，相同的值域，定义域有 $9$ 种可能。

但是题中要求定义域不同，而上面 $9$ 种定义域有一种其实就是题中提到的。因此，答案为 $8$。

已知函数 $f(x)$ 的定义域为 {$1,2,3,4$}，值域为 {$7,8,9$}，且对任意的 $x<y$ 都有 $f(x)\le f(y)$，则满足条件的不同函数共有多少个？

值域中每一个数，都至少有一个定义域中的数与之对应；而定义域中的每一个数，都恰好对应一个值域中的数。分析题目，题中要求定义域中有 $4$ 个数，可值域只有 $3$ 个数，换句话说，恰好有 $2$ 个定义域的数，对应的值域中的数，是同一个。

而题中还说 $x<y,f(x)\le f(y)$，结合上文，我们可以列出不等式：

这个式子中的 $3$ 个不等号有且只有 $1$ 个实际上是取等号的。如果有大于 $1$ 个不等号实际上取等号，就会导致函数可能的值的数量小于 $3$，值域中有数找不到定义域中的数与之对应；而如果没有不等号实际上取等号，就会导致 $4$ 个 $x$ 产生了 $4$ 个 $f(x)$，可值域的大小才 $3$ 个。

而任何一个不等号实际上取了等号，我们都能构造出对应关系。

因此，满足条件的不同函数，有 $3$ 个。

2、与一元二次函数结合起来

前置知识

已知函数 $f(x)=x^2-4x$ 在 $[0,m]$ 上的值域为 $[-4,0]$，则实数 $m$ 的取值范围是？

首先，通过图象，我们可以看出 $x^2-4x$ 的情况：

可以计算得出，当 $x=0$ 时，函数值为 $0$。当 $x=2$ 时，函数值取到最小值，为 $-4$。当 $x=4$ 时，函数值再次为 $0$。

因此，可以得出 $2\le m\le4$。如果 $m$ 太小，函数值就不够小；如果 $m$ 太大，函数值就会超过 $0$。

3、当给出的函数不标准的时候

有些问题给出的函数，不会给 $f(x)$ 这种标准的形式，而是各种各样乱七八糟的形式。

已知 $f(2x-1)=4x^2$，则下列结论中正确的是：

A. $f(3)=-9$

B. $f(-3)=-4$

C. $f(x)=x^2$

D. $f(x)=(x+1)^2$

对于这类问题，首先为了避免混淆，先将题干中给出的函数变量改个名字，变成 $f(2a-1)=4a^2$。

然后，对于每个选项，挨个儿解出函数括号内的数对应的 $a$ 是多少。

例如，A 选项，我们可得 $2a-1=3$，解得 $a=2,4a^2=16\neq-9$。A 选项错误。

B 选项，我们可得 $2a-1=-3$，解得 $a=-1,4a^2=4\neq-4$，B 选项错误。

C 和 D 选项，我们可得 $2a-1=x$，稍稍细心计算，得 $a=\frac{x+1}{2}$，由此可得 $4a^2=(x+1)^2$，D 选项正确而 C 选项错误。

对于此类问题，应将题干中给出的原函数传入的参数（即 $2a-1$）和选项中给出的待验证函数传入的参数（即 $3$、$-3$、$x$）进行比对，要求两者相等，得出原函数传入的参数中的字母具体是什么（即，当 $2a-1=3$ 时 $a=2$），然后将字母代入计算。为避免混淆，可以将原函数中的字母统一更名。

4、仔细分析更复杂的函数

设 $f(x)=\frac{1+x^2}{1-x^2}$，则下列结论正确的是？请注意本题多选。

A. $f(-x)=-f(x)$

B. $f(\frac{1}{x})=-f(x)$

C. $f(-\frac{1}{x})=f(x)$

D. $f(-x)=f(x)$

我们可以发现，$x^2=(-x)^2$，$x$ 的正负性并不影响平方的结果。因此，$\frac{1+x^2}{1-x^2}=\frac{1+(-x)^2}{1-(-x)^2}$，由此可得 $f(-x)=f(x)$，A 选项错误，D 选项正确。

顺便说一句，这样的函数叫做偶函数，函数关于 $y$ 轴对称，图象如下：

同时，我们可以得到 $f(\frac{1}{x})=f(-\frac{1}{x})$，这一条待会有用。

接下来，细心推出以下式子：

$f(\frac{1}{x})\\=\frac{1+\frac{1}{x}^2}{1-\frac{1}{x}^2}\\=\frac{x^2+1}{x^2}\div\frac{x^2-1}{x^2}\\=\frac{1+x^2}{x^2-1}$

至此，函数 $f(\frac{1}{x})$ 计算完毕。

而函数 $f(x)=\frac{1+x^2}{1-x^2}$，可以很容易地发现 $f(\frac{1}{x})=-f(x)$，B 选项正确，C 选项错误。图象如下：

可以恰好发现 $f(x)$ 与 $f(\frac{1}{x})$ 关于 $x$ 轴对称。

对于此类问题，应细心审题，谨慎作答，答后验证。

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