{"id":2189,"date":"2024-08-23T20:39:45","date_gmt":"2024-08-23T12:39:45","guid":{"rendered":"http:\/\/www.jiaohuweike.online\/?p=2189"},"modified":"2024-08-23T20:39:45","modified_gmt":"2024-08-23T12:39:45","slug":"hsyldwt-5-4-3","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=2189","title":{"rendered":"\u51fd\u6570\u7684\u6027\u8d28"},"content":{"rendered":"\n<div class=\"wp-block-stackable-spacer stk-block-spacer stk--no-padding stk-block stk-959acda\" data-block-id=\"959acda\"><\/div>\n\n\n\n<div class=\"wp-block-ht-block-toc is-style-rounded htoc htoc--position-wide toc-list-style-bulleted\" data-htoc-state=\"expanded\"><span class=\"htoc__title\"><span class=\"ht_toc_title\">\u5185\u5bb9\u76ee\u5f55<\/span><span class=\"htoc__toggle\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\"><g fill=\"#444\"><path d=\"M15 7H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1z\"><\/path><path d=\"M15 1H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1zM15 13H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1z\"><\/path><\/g><\/svg><\/span><\/span><div class=\"htoc__itemswrap\"><ul class=\"ht_toc_list\"><li class=\"\"><a href=\"\/#htoc-\">\u51fd\u6570\u7684\u5355\u8c03\u6027<\/a><\/li><li class=\"\"><a href=\"\/#htoc-11111111111111111\">\u51fd\u6570\u7684\u5947\u5076\u6027\u3001\u5468\u671f\u6027<\/a><\/li><li class=\"\"><a href=\"\/#htoc-1111111111111111111111111111111\">\u51fd\u6570\u7684\u5bf9\u79f0\u6027<\/a><\/li><li class=\"\"><a href=\"\/#htoc-111111111111111111111111111111111111111\">\u5bfc\u51fd\u6570\u4e0e\u539f\u51fd\u6570\u7684\u5bf9\u79f0\u6027<\/a><\/li><\/ul><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-\">\u51fd\u6570\u7684\u5355\u8c03\u6027<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.\uff08\u591a\u9009\uff09\u9009\u51fa\u6b63\u786e\u7684\u9009\u9879(~~~~~~)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\nA.\u5bf9\u4e8e\u51fd\u6570  y=f(x) , \u82e5  f(1)&lt; f(3) , \u5219  f(x)  \u4e3a  (1,3)  \u7684\u589e\u51fd\u6570.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nB.\u51fd\u6570  f(x)  \u7684\u5b9a\u4e49\u57df\u4e3a  I , \u533a\u95f4  D \\subseteq I , \u82e5  \\forall x_{1}, x_{2} \\in D , \u5f53  x_{1}&lt; x_{2}  \u65f6, \u90fd\u6709  f\\left(x_{1}\\right)&lt; f\\left(x_{2}\\right) ,\\\\\u90a3\u4e48\u5c31\u79f0\u51fd\u6570  f(x)  \u5728\u533a\u95f4  D  \u4e0a\u5355\u8c03\u9012\u589e.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nC.\u51fd\u6570  f(x), x \\in D , \u82e5\u5bf9\u4efb\u610f  x_{1}, x_{2} \\in D , \u4e14  x_{1} \\neq x_{2}  \u6709  \\left(x_{1}-x_{2}\\right)\\left[f\\left(x_{1}\\right)-f\\left(x_{2}\\right)\\right]&gt;0 ,~~~~~\\\\\u5219\u51fd\u6570  f(x)  \u5728\u533a\u95f4  D  \u4e0a\u662f\u589e\u51fd\u6570.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nD.\u51fd\u6570  f(x), x \\in D , \u82e5\u5bf9\u4efb\u610f  x_{1}, x_{2} \\in D , \u4e14  x_{1} \\neq x_{2},  \u6709 \\frac{f(x_1)-f(x_2)}{x_1-x_2}&lt;0,\u5219\u51fd\u6570  f(x)~~  \\\\\u5728\u533a\u95f4  D  \u4e0a\u662f\u589e\u51fd\u6570.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nE.\u51fd\u6570  f(x), g(x)  \u7684\u5b9a\u4e49\u57df\u90fd\u4e3a  \\mathbf{R} , \u4e14  f(x)&gt;0, g(x)&gt;0, f(x)  \u662f\u589e\u51fd\u6570,  g(x)  \u662f\u589e\u51fd\u6570, ~~\\\\\u5219  g(x)+f(x)  \u662f\u589e\u51fd\u6570,  f(x) g(x)  \u4e5f\u662f\u589e\u51fd\u6570.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-bce-e-f-x-g-x-gt-0\"><pre>\u7b54\u6848\uff1aBE.\u63d0\u793a\uff1aE\u4e2d{[f(x) g(x)]}' &gt;0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.\u753b\u4e0b\u5217\u51fd\u6570\u7684\u56fe\u8c61\uff0c\u5e76\u6c42\u5355\u8c03\u533a\u95f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(1)y=|x-2|;(2)y=\\frac{-2}{x-1}+1;(3)y=|x^2+2x-3|;(4)y=x^2+2|x|-3;~~~~~~~~~~~~~~~~~~~\\\\(5)y=x-\\frac{1}{x};(6)y=\\frac{\\ln x}{x};(7)y=\\frac{x}{1+|x|}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-4-7-x-gt-0-f-x-y-frac-x-1-x-1-frac-1-1-x\"><pre>\u63d0\u793a\uff1a\u9898(4)\u4e3a\u5076\u51fd\u6570\uff1b\u9898(7)\u4e3a\u5947\u51fd\u6570\uff0cx&gt;0\u65f6\uff0cf(x)=y=\\frac{x}{1+x}=1-\\frac{1}{1+x}.~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-ada044f\" data-block-id=\"ada044f\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-ada044f-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-2cceaba\" data-v=\"4\" data-block-id=\"2cceaba\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-2cceaba-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-2cceaba-inner-blocks\">\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/09\/jj1.png\" alt=\"\" class=\"wp-image-2655\" style=\"width:198px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-ad7c4cc\" data-v=\"4\" data-block-id=\"ad7c4cc\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-ad7c4cc-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-ad7c4cc-inner-blocks\">\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/09\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-08-23-210637.png\" alt=\"\" class=\"wp-image-2657\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-69f1dac\" data-v=\"4\" data-block-id=\"69f1dac\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-69f1dac-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-69f1dac-inner-blocks\">\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/09\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-08-23-210734.png\" alt=\"\" class=\"wp-image-2658\" style=\"width:179px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:100%\">\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-07219cd\" data-block-id=\"07219cd\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-07219cd-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-c6a264d\" data-v=\"4\" data-block-id=\"c6a264d\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-c6a264d-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-c6a264d-inner-blocks\">\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/09\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-08-23-210908.png\" alt=\"\" class=\"wp-image-2659\" style=\"width:166px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-48f07f6\" data-v=\"4\" data-block-id=\"48f07f6\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-48f07f6-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-48f07f6-inner-blocks\">\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/09\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-08-23-210954.png\" alt=\"\" class=\"wp-image-2660\" style=\"width:210px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-cc1f077\" data-v=\"4\" data-block-id=\"cc1f077\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-cc1f077-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-cc1f077-inner-blocks\">\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/08\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-08-23-211044-300x251-1.png\" alt=\"\" class=\"wp-image-2662\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-7349c76\" data-block-id=\"7349c76\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-7349c76-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-0339f26\" data-v=\"4\" data-block-id=\"0339f26\"><style>@media screen and (min-width:690px){.stk-0339f26 {flex:var(--stk-flex-grow, 1) 1 calc(33.33400000000002\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/08\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-09-05-163811-1.png\" alt=\"\" class=\"wp-image-2663\"\/><\/figure>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-96b87f0\" data-v=\"4\" data-block-id=\"96b87f0\"><style>@media screen and (min-width:690px){.stk-96b87f0 {flex:var(--stk-flex-grow, 1) 1 calc(66.666\n<\/div><\/div>\n<\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.(1)\u6c42\u51fd\u6570y=\\log_{0.5}(|x|-1)\u7684\u5355\u8c03\u9012\u51cf\u533a\u95f4;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)y=\\log _{\\frac{1}{3}}\\left(-x^{2}+4 x-3\\right)  \u7684\u5355\u8c03\u9012\u589e\u533a\u95f4\uff1b~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(3)\u5df2\u77e5\u51fd\u6570  f(x)=\\ln \\left(x^{2}-3 x-4\\right)  \u5728  (a,+\\infty)  \u4e0a\u5355\u8c03\u9012\u589e, \u6c42  a  \u7684\u53d6\u503c\u8303\u56f4.~~~~~~~~~~~~~~~~~~~\\\\\n(4)\u5df2\u77e5  f(x)=\\ln \\left(x^{2}-a x+2 a-2\\right)(a&gt;0) , \u82e5  f(x)  \u5728  [1,2)  \u4e0a\u5355\u8c03, \u6c42 a  \u7684\u8303\u56f4.~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-1-infty-2-2-3\"><pre>\u7b54\u6848\uff1a(1)(1,+\\infty);(2)[2,3)\u63d0\u793a\uff1a\u5148\u6c42\u5b9a\u4e49\u57df\uff0c\u6839\u636e\u590d\u5408\u51fd\u6570\u5355\u8c03\u6027;(3)[4,+\\infty);~~~~~~~\\\\(4)(1,2] \\cup[4,+\\infty) ;\u63d0\u793a\uff1a\u5bf9\u79f0\u8f74x=\\frac{a}{2},\u4ee4g(x)=x^{2}-a x+2 a-2,\u53ea\u9700\u6ee1\u8db3~~~~~~~~~~~~~\\\\ \\left\\{\\begin{matrix}\n 0&lt; \\displaystyle\\frac{a}{2}\\le 1  &amp; \\\\\n g(1)&gt; 0 &amp;\n\\end{matrix}\\right.\u6216\\left\\{\\begin{matrix}\n  \\displaystyle\\frac{a}{2}\\ge 2  &amp; \\\\\n g(2)\\ge 0 &amp;\n\\end{matrix}\\right.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.\u5df2\u77e5f(x)=x^2-2x-3,g(x)=f(5-x^2),\u6c42g(x)\u7684\u5355\u8c03\u533a\u95f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-2-0-2-infty-infty-2-0-2-g-x-f-t-t-5-x-2-f-t-t-lt-1-f-t-t-gt-1-f-t-t-5-x-2-t-gt-1-t-lt-1\"><pre>\u7b54\u6848\uff1a\u5355\u589e\u533a\u95f4\u4e3a(-2,0),(2,+\\infty);\u5355\u51cf\u533a\u95f4\u4e3a(-\\infty\uff0c-2),(0,2).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u63d0\u793a\uff1a\u590d\u5408\u51fd\u6570\u8868\u793a\u4e3ag(x)=f(t),t=5-x^2,\u5bf9\u4e8ef(t),t&lt;1\u65f6\uff0cf(t)\u51cf\uff0ct&gt;1\u65f6\uff0c~~\\\\f(t)\u589e\uff0c\u51fd\u6570t=5-x^2\u5728t&gt;1\u548ct&lt;1\u65f6\u7684\u5355\u8c03\u6027\u662f\u663e\u7136\u7684\uff0c\u7531\u590d\u5408\u51fd\u6570\u5355\u8c03\u6027\u53ef\u89e3.<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5.\u5df2\u77e5\u51fd\u6570y=9^x+m\\cdot 3^x-3\u5728\u533a\u95f4[-2,2]\u4e0a\u5355\u8c03\u9012\u51cf\uff0c\u6c42m\u7684\u53d6\u503c\u8303\u56f4.~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-infty-18-t-3-x-therefore-t-in-frac-1-9-9-t-in-frac-1-9-9-y-t-2-mt-3-therefore-x-frac-m-2-ge9\"><pre>\u7b54\u6848\uff1a(-\\infty,-18];\u63d0\u793a\uff1a\u4ee4t=3^x,\\therefore t\\in [\\frac{1}{9},9]\uff0c\u4e14\u5355\u589e\u3002t\\in [\\frac{1}{9},9]\u5e94\u5728\u51fd\u6570~~~~~~~~~~~~~~~~~\\\\y=t^2+mt-3\u7684\u5355\u8c03\u9012\u51cf\u533a\u95f4\u4e0a\uff0c\\therefore\u5bf9\u79f0\u8f74x=-\\frac{m}{2}\\ge9.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6.\u51fd\u6570f(x)\u662fR\u4e0a\u7684\u51cf\u51fd\u6570\uff0c\u82e5y=f(ax^2-2x)\u5728(1,+\\infty)\u4e0a\u662f\u589e\u51fd\u6570\uff0c\u6c42a\u7684\u8303\u56f4.~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-a-le-0-t-ax-2-2x-1-infty-a-0-a-gt-0-a-lt-0\"><pre>\u7b54\u6848\uff1aa\\le 0;\u63d0\u793a\uff1a\u53ea\u9700t=ax^2-2x\u5728(1,+\\infty)\u4e0a\u662f\u51cf\u51fd\u6570\uff0c\u8ba8\u8bbaa=0,a&gt;0,a&lt;0.<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>7.\u51fd\u6570  y=f(x)  \u662f\u5b9a\u4e49\u5728  [-2,2]  \u4e0a\u7684\u51cf\u51fd\u6570, \u4e14  f(a+1)&lt; f(2 a) , \u6c42\u5b9e\u6570  a  \u7684\u53d6\u503c\u8303\u56f4.\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-1\"><pre>\u7b54\u6848:[-1,1) ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>8.\u5df2\u77e5\u51fd\u6570  f(x)=\\ln x+2^{x} , \u82e5  f\\left(x^{2}-4\\right)&lt;2 ,\u6c42\u5b9e\u6570  x  \u7684\u53d6\u503c\u8303\u56f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-sqrt-5-2-cup-2-sqrt-5-f-left-x-2-4-right-lt-f-1\"><pre>\u7b54\u6848:(-\\sqrt{5},-2) \\cup(2, \\sqrt{5}) ;\u63d0\u793a\uff1a\u4e0d\u7b49\u5f0f\u8f6c\u5316\u4e3a f\\left(x^{2}-4\\right)&lt;f(1).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>9.\u5df2\u77e5\u51fd\u6570  f(x)=\\left\\{\\begin{array}{l}-x^{2}-2 x, x \\geqslant 0, \\\\ x^{2}-2 x, x&lt;0,\\end{array}\\right.  \u82e5  f\\left(3-a^{2}\\right)&lt; f(2 a) , \u6c42\u5b9e\u6570  a  \u7684\u53d6\u503c\u8303\u56f4.~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-3-1-f-x-r\"><pre>\u7b54\u6848\uff1a(-3,1);\u63d0\u793a\uff1af(x)\u5728R\u4e0a\u5355\u8c03\u9012\u51cf.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>10.\u5df2\u77e5\u51fd\u6570  f(x+2)  \u662f  \\mathbf{R}  \u4e0a\u7684\u5076\u51fd\u6570, \u4e14  f(x)  \u5728  [2,+\\infty)  \u4e0a\u6052\u6709  \\frac{f\\left(x_{1}\\right)-f\\left(x_{2}\\right)}{x_{1}-x_{2}}&lt;0~~~~~~~~~~~\\\\\\left(x_{1} \\neq x_{2}\\right) , \u6c42\u4e0d\u7b49\u5f0f  f(\\ln x)&gt;f(1)  \u7684\u89e3\u96c6.~~~~~~~~~~~~~~~~~\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-e-e-3-f-x-2-mathbf-r-x-2-therefore-ln-x-2-lt-1-2-therefore-x-in-e-e-3\"><pre>\u7b54\u6848\uff1a(e,e^3);\u63d0\u793a\uff1a\u7531 f(x+2)  \u662f  \\mathbf{R}  \u4e0a\u7684\u5076\u51fd\u6570\u53ef\u5f97\u5bf9\u79f0\u8f74\u4e3ax=2,\u53c8\u51fd\u6570\u5de6\u589e\u53f3\u51cf,\\\\\\therefore |\\ln x-2|&lt;|1-2|,\\therefore x\\in(e,e^3).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>11.\u8bbe\u51fd\u6570  f(x)=\\left\\{\\begin{array}{l}a x-1, x &lt; a, \\\\ x^{2}-2 a x+1, x \\geqslant a,\\end{array}\\right.  \u5f53  f(x)  \u5b58\u5728\u6700\u5c0f\u503c\u65f6, \u6c42\u5b9e\u6570  a \u7684\u53d6\u503c\u8303\u56f4.~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-a-le-1-a-0\"><pre>\u7b54\u6848\uff1aa\\le -1\u6216a=0;\u63d0\u793a\uff1a\u753b\u51fd\u6570\u56fe\u8c61.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>12.\u5df2\u77e5\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u7684\u51fd\u6570  f(x)  \u6ee1\u8db3: (1)  f(x+y)=f(x)+f(y)+1 , (2) \u5f53  x&gt;0  \u65f6,~~~~~~~  \\\\f(x)&gt;  -1 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(1) \u6c42  f(0)  \u7684\u503c, \u5e76\u8bc1\u660e  f(x)  \u5728  \\mathbf{R}  \u4e0a\u662f\u589e\u51fd\u6570;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2) \u82e5  f(1)=1 , \u89e3\u5173\u4e8e  x  \u7684\u4e0d\u7b49\u5f0f  f\\left(x^{2}+2 x\\right)+f(1-x)&gt;4 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-f-0-1-2-infty-2-cup-1-infty-f-left-x-2-2-x-right-f-1-x-f-x-2-2-x-1-x-1-gt-4-therefore-f-x-2-2x-1-gt-5-f-3-5-therefore-f-x-2-2x-1-gt-f-3\"><pre>\u7b54\u6848\uff1a(1)f(0)=-1;(2)(-\\infty,-2)\\cup (1,+\\infty);\u63d0\u793a\uff1a f\\left(x^{2}+2 x\\right)+f(1-x)=~~~~~~~~~~~~~\\\\f(x^{2}+2 x+1-x)-1&gt;4,\\therefore  f(x^{2}+2x+1)&gt;5,\u53c8f(3)=5,\\therefore  f(x^{2}+2x+1)&gt;f(3)<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>13.\u5df2\u77e5\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u7684\u5076\u51fd\u6570  f(x)  \u5728  [0,+\\infty)  \u4e0a\u5355\u8c03\u9012\u589e, \u82e5  f(\\ln x)&lt; f(2) , \u6c42  x  \u7684\u8303\u56f4.~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-left-mathrm-e-2-mathrm-e-2-right-ln-x-lt-2\"><pre>\u7b54\u6848\uff1a\\left(\\mathrm{e}^{-2}, \\mathrm{e}^{2}\\right) ;\u63d0\u793a\uff1a\u89c2\u5bdf\u79bb\u5bf9\u79f0\u8f74\u7684\u8ddd\u79bb\uff0c\u5f97|\\ln x|&lt;2.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>14.\u8bbe\u51fd\u6570  f(x)=\\ln (1+|x|)-\\frac{1}{1+x^{2}} , \u6c42\u4f7f\u5f97  f(x)&gt;f(2 x-1) \u6210\u7acb\u7684  x  \u7684\u53d6\u503c\u8303\u56f4\u4e3a.<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-left-frac-1-3-1-right-f-x-0-infty-therefore-x-gt-2x-1\"><pre>\u7b54\u6848\uff1a\\left(\\frac{1}{3}, 1\\right) ;\u63d0\u793a\uff1a\u51fd\u6570f(x)\u4e3a\u5076\u51fd\u6570\uff0c\u5728(0,+\\infty)\u5355\u589e\uff0c\\therefore |x|&gt;|2x-1|.~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>15.\u5df2\u77e5\u51fd\u6570  f(x)=\\frac{3^{x}-1}{3^{x}+1}+3 x  +3 , \u4e14  f\\left(a^{2}\\right)+f(3 a-4)&gt;6 , \u6c42\u5b9e\u6570  a  \u7684\u53d6\u503c\u8303\u56f4.~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-infty-4-cup-1-infty-g-x-f-x-3-frac-3-x-1-3-x-1-3-x-f-left-a-2-right-f-3-a-4-gt-6-f-a-2-3-gt-f-3a-4-3-g-a-2-gt-g-3a-4\"><pre>\u7b54\u6848\uff1a(-\\infty,-4)\\cup(1,+\\infty);\u63d0\u793a\uff1a\u4ee4g(x)=f(x)-3=\\frac{3^{x}-1}{3^{x}+1}+3 x \u662f\u5947\u51fd\u6570\uff0c\u4e14~~~~~\\\\\u5355\u589e, f\\left(a^{2}\\right)+f(3 a-4)&gt;6\u53d8\u5f62\u4e3af(a^2)-3&gt;-[f(3a-4)-3]\uff0c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u5373g(a^2)&gt;-g(3a-4).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u7ec3\u4e601.\u5df2\u77e5f(x)=e^{-x}-e^x-2x+4,\u82e5f(a-6)+f(a^2)&gt;8,\u6c42a\u7684\u8303\u56f4.~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-3-2-f-x-f-x-4-f-a-6-4-gt-f-a-2-4\"><pre>\u7b54\u6848:(-3,2);\u63d0\u793a\uff1aF(x)=f(x)-4\u4e3a\u5947\u51fd\u6570\uff0c\u4e14\u5355\u51cf\u51fd\u6570\u3002\u539f\u5f0f\u53d8\u5f62\u4e3a~~~~~~~~~~~~~~~~~~~\\\\f(a-6)-4&gt;-f(a^2)+4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u7ec3\u4e602. \u5df2\u77e5\u51fd\u6570  f(x)=-2 x^{3}+\\frac{c}{\\mathrm{e}^{x}+1}  \u7684\u56fe\u8c61\u5173\u4e8e\u70b9  (\\mathbf{0}, \\mathbf{1})  \u6210\u4e2d\u5fc3\u5bf9\u79f0\u56fe\u5f62,  f\\left(-t^{2}\\right)+~~~\\\\f(2 t+3)&gt;2 , \u6c42\u5b9e\u6570  t  \u7684\u53d6\u503c\u8303\u56f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-da11\"><pre>\u7b54\u6848\uff1a\\left ( -\\infty ,-1 \\right ) \\cup \\left ( 3,+\\infty  \\right ) ;\u63d0\u793a:\u5173\u4e8e(0,1)\u5bf9\u79f0\uff0c\\Rightarrow f(x)+f(-x)=2,\\Rightarrow c=2,~~\\\\\u6784\u9020F(x)=f(x)-1\uff0cF(x)\u5173\u4e8e(0,0)\u5bf9\u79f0\uff0c\\therefore F(x)\u4e3a\u5947\u51fd\u6570\uff0c\u4e14\u4e3a\u51cf\u51fd\u6570,\u7531~~~~~~~~~~~\\\\f\\left(-t^{2}\\right)+f(2 t+3)&gt;2\\Rightarrow f\\left(-t^{2}\\right)-1&gt;-[f(2 t+3)-1],\\Rightarrow F(-t^2)&gt;-F(2t+3).<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u7ec3\u4e603.\u5b9a\u4e49\u5728  (0,+\\infty)  \u4e0a\u7684\u51fd\u6570  f(x)  \u6ee1\u8db3  \\forall x_{1}, x_{2} \\in(0,+\\infty)  \u4e14  x_{1} \\neq x_{2} , \u6709~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\  \\left[f\\left(x_{1}\\right)-f\\left(x_{2}\\right)\\right]\\left(x_{1}-x_{2}\\right)&gt;0 , \u4e14  f(x y)=f(x)+f(y), f(4)=\\frac{2}{3} , \u6c42\u4e0d\u7b49\u5f0f ~~~~~~~~~~~~~~~~~~~~~ \\\\f(2 x)-f(x-3)&gt;1  \u7684\u89e3\u96c6.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-da111\"><pre>\u7b54\u6848:(3,4);\u63d0\u793a:x=y=2,\\Rightarrow f(2)=\\frac{1}{3} ,\\Rightarrow f(8)=1.f(2 x)-f(x-3)&gt;1  \\Leftrightarrow ~~~~~~~~~~\\\\f(2 x) &gt; f(x-3)+f(8),\\Rightarrow f(2 x)&gt;f(8x-24),\u7ed3\u5408\u5b9a\u4e49\u57df\\Rightarrow  x\\in(3,4).~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>16.\u5df2\u77e5\u51fd\u6570  f(x)=\\left\\{\\begin{array}{l}2-a x, x \\leq 1 \\\\ \\displaystyle \\frac{1}{3} x^{3}-\\frac{3}{2} a x^{2}+\\left(2 a^{2}+2\\right) x-\\frac{11}{6}, x&gt;1\\end{array}\\right. , \u82e5\u5bf9\u4efb\u610f  x_{1}&lt; x_{2} ,~~~~~~~~ \\\\\u90fd\u6709  f\\left(x_{1}\\right)-f\\left(x_{2}\\right)&lt;2 x_{1}-2 x_{2} , \u5219\u5b9e\u6570  a  \u7684\u53d6\u503c\u8303\u56f4\u662f ( ~~)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  (-\\infty,-2) ~~\nB.  [1,+\\infty) ~~\nC.  \\left(-2, \\frac{1}{2}\\right] ~~\nD.  \\left(-\\infty,-\\frac{3}{4}\\right] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-da\"><pre>\u7b54\u6848\uff1aA;\u63d0\u793a\uff1af\\left(x_{1}\\right)-f\\left(x_{2}\\right)&lt;2 x_{1}-2 x_{2} ,\u8f6c\u5316\u4e3af\\left(x_{1}\\right)-2 x_{1}&lt; f\\left(x_{2}\\right)-2 x_{2} ,~~~~~~~~~~~\\\\\u5373g(x)=f(x)-2x\u5728R\u4e0a\u5355\u8c03\u9012\u589e\uff1b~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-11111111111111111\">\u51fd\u6570\u7684\u5947\u5076\u6027\u3001\u5468\u671f\u6027<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u5e38\u89c1\u7ed3\u8bba<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(1) \u82e5  f(x+a)=f(x) , \u5219  T=a \uff1b(2)  \u82e5  f(x+a)=-f(x) , \u5219  T=2 a;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(3) \u82e5  f(x+a)=\\frac{1}{f(x)} , \u5219  T=2 a\uff1b(4) \u82e5  f(x+a)=\\frac{m}{f(x)} , \u5219  T=2 a(m\\ne 0) ;~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(5) \u82e5  f(x+a)=-\\frac{1}{f(x)} , \u5219  T=2 a.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.(\u591a\u9009)\u4e0b\u5217\u547d\u9898\u4e2d\u6b63\u786e\u7684\u662f( ~~~~~)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  f(x)=\\log _{2}\\left(x+\\sqrt{x^{2}+1}\\right)  \u662f\u5947\u51fd\u6570~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nB. \u51fd\u6570  y=x \\sin x  \u662f\u5076\u51fd\u6570~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nC.  f(x)=(x-1) \\sqrt{\\frac{1+x}{1-x}}  \u662f\u5076\u51fd\u6570~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\ D. f(x)=\\left\\{\\begin{array}{ll}x^{2}+x &amp; (x&lt;0), \\\\ -x^{2}+x &amp; (x&gt;0)\\end{array}\\right.  \u662f\u5947\u51fd\u6570 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\E.  f(x)=\\frac{\\sqrt{1-x^{2}}}{|x+3|-3}  \u662f\u5947\u51fd\u6570~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\F.  f(x)=\\frac{\\sqrt{36-x^{2}}}{|x+3|-3}  \u662f\u5947\u51fd\u6570~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\G.  f(x)=\\frac{a^x-1}{a^x+1}(a&gt;0,a\\ne 1)  \u662f\u5947\u51fd\u6570~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-abdeg-c-1-1-e-1-1-f-x-frac-sqrt-1-x-2-x-e-6-6\"><pre>\u7b54\u6848\uff1aABDEG;\u63d0\u793a\uff1a\u9009\u9879C\u7684\u5b9a\u4e49\u57df\u4e3a[-1,1)\uff0c\u4e0d\u5bf9\u79f0\uff1b\u9009\u9879E\u7684\u5b9a\u4e49\u57df\u4e3a[-1,1],\\\\\u5316\u7b80\u540e\u4e3af(x)=\\frac{\\sqrt{1-x^{2}}}{x} ;\u9009\u9879E\u7684\u5b9a\u4e49\u57df\u4e3a(-6,6]\u4e0d\u5bf9\u79f0\uff0c\u975e\u5947\u975e\u5076\u51fd\u6570\u3002~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.\u82e5  f(x)=(x+a) \\ln \\frac{2 x-1}{2 x+1}  \u4e3a\u5076\u51fd\u6570\uff0c\u6c42\u5b9e\u6570  a.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-a-0\"><pre>\u7b54\u6848\uff1aa=0.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\u82e5  f(x)=\\frac{a+1}{\\mathrm{e}^{x}-1}+1  \u4e3a\u5947\u51fd\u6570, \u6c42\u5b9e\u6570  a.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-a-1\"><pre>\u7b54\u6848\uff1aa=1.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.\u5df2\u77e5\u51fd\u6570  f(x)  \u4e3a\u5947\u51fd\u6570\u4e14\u5b9a\u4e49\u57df\u4e3a  \\mathbf{R} , \u5f53  x&gt;0  \u65f6,  f(x)=x+1 , \u6c42f(x)\u7684\u89e3\u6790\u5f0f.~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-f-x-left-begin-matrix-x-1-x-gt-0-amp-0-x-0-amp-x-1-x-lt-0-amp-end-matrix-right\"><pre>\u7b54\u6848\uff1af(x)=\\left\\{\\begin{matrix}\nx+1,x&gt;0,  &amp; \\\\\n 0,x=0, &amp; \\\\\nx-1,x&lt;0. &amp;\n\\end{matrix}\\right.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u53d8\u5f0f.\u82e5\u51fd\u6570g(x)\u4e0ef(x)=e^x\u5173\u4e8e\u76f4\u7ebfx=1\u5bf9\u79f0,\u6c42g(x)\u7684\u89e3\u6790\u5f0f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-g-x-2-2-x-x-y-g-x-x-1-2-x-y-f-x-y-e-2-x-therefore-g-x-e-2-x\"><pre>\u7b54\u6848\uff1ag(x)=2^{2-x};\u63d0\u793a\uff1a\u8bbe(x,y)\u662fg(x)\u4e0a\u4e00\u70b9\uff0c\u5173\u4e8ex=1\u7684\u5bf9\u79f0\u70b9\u4e3a(2-x,y),\u4e14~~\\\\\u5728f(x)\u56fe\u8c61\u4e0a\uff0c\u5373y=e^{2-x},\\therefore g(x)=e^{2-x}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5.\u8bbe  f(x)  \u662f\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u5468\u671f\u4e3a 4 \u7684\u5076\u51fd\u6570, \u4e14\u5f53  x \\in[0,2]  \u65f6,  f(x)=\\log _{2}(x+1) ,\u6c42\u51fd~~~~~~~\\\\\u6570  f(x)  \u5728  [2,4]  \u4e0a\u7684\u89e3\u6790\u5f0f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-f-x-log-2-5-x-x-in-2-4-x-in-2-4-x-4-in-2-0-4-x-in-0-2-therefore-f-4-x-log-2-5-x-f-4-x-f-x-f-x-therefore-f-x-log-2-5-x-x-in-2-4\"><pre>\u7b54\u6848\uff1af(x)=\\log_2(5-x),x\\in[2,4];\u63d0\u793a\uff1a\u8bbex\\in[2,4],\u5219x-4\\in[-2,0],~~~~~~~~~~~~~~~~~~~~~~\\\\\u52194-x\\in  [0,2],\\therefore f(4-x)=\\log_2(5-x),f(4-x)=f(-x)=f(x),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\\therefore f(x)=\\log_2(5-x),x\\in[2,4].~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6. f(x)  \u662f\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u7684\u5076\u51fd\u6570, \u5bf9  \\forall x \\in \\mathbf{R} , \u5747\u6709  f(x+2)=-f(x) , \u5f53  x \\in[0,1]  \u65f6,  ~~~~~~~~~~~~~\\\\f(x)=\\log _{2}(2-x) ,\n\u5f53  x \\in[2,3]  \u65f6, \u6c42f(x)\u7684\u89e3\u6790\u5f0f. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-f-x-log-2-4-x-x-in-2-3-x-in-2-3-x-2-in-0-1-therefore-f-x-2-log-2-4-x-f-x-2-f-x-therefore-f-x-log-2-4-x\"><pre>\u7b54\u6848\uff1af(x)=-\\log _{2}(4-x) \uff0c x \\in[2,3] ;\u63d0\u793a\uff1a\u8bbe x \\in[2,3]\uff0c\u5219x-2\\in [0,1],~~~~~~~~~~~~~~~~~\\\\\\therefore f(x-2)=\\log _{2}(4-x) ,\u7531\u6761\u4ef6\u77e5 f(x-2)=-f(x),\\therefore f(x)=-\\log _{2}(4-x)~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>7.\uff08\u591a\u9009\uff09\u5df2\u77e5\u51fd\u6570  f(x)  \u7684\u5b9a\u4e49\u57df\u4e3a  \\mathbf{R} , \u6ee1\u8db3  f(x+3)+f(x+1)=0 , \u4e14  f(x+1)  \u4e3a\u5076~~\\\\\u51fd\u6570, \u5219 (~ )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  f(2)=0 ~~~\nB.  f(x)  \u4e3a\u5076\u51fd\u6570~~~\nC.  f(x)  \u4e3a\u5468\u671f\u51fd\u6570~~~\nD.  f(x+4)  \u4e3a\u5076\u51fd\u6570~~~~~~~~~~~~~~~~~~~~~~~~~\n\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-ac-f-x-1-f-x-1-f-x-1-x-1-f-x-3-f-x-1-0-f-x-3-f-x-1-f-x-1-t-4-2-0\"><pre>\u7b54\u6848\uff1aAC\uff1b\u63d0\u793a\uff1a \u7531f(x+1)  \u4e3a\u5076\u51fd\u6570\u53ef\u5f97\u4ee5\u4e0b\u4e24\u4e2a\u7ed3\u8bba\uff1af(-x+1)=f(x+1)\u548c\\\\x=1\u4e3a\u5bf9\u79f0\u8f74.  \u7531f(x+3)+f(x+1)=0\u53ef\u5316\u4e3af(x+3)=-f(x+1)=-f(-x+1),~~\\\\\u53ef\u5f97\u4e24\u4e2a\u7ed3\u8bba\uff1a\u5468\u671fT=4\u548c(2,0)\u4e3a\u5bf9\u79f0\u4e2d\u5fc3.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>8.\u5df2\u77e5\u51fd\u6570f(x)\u5b9a\u4e49\u57df\u4e3aR,(f(x)\u4e0d\u6052\u4e3a0),f(x+2)\u4e3a\u5076\u51fd\u6570\uff0cf(2x+1)\u4e3a\u5947\u51fd\u6570\uff0c\\\\\u6c42f(-1).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-f-1-0-f-x-2-rightarrow-x-2-f-2x-1-rightarrow-f-2x-1-f-2x-1-rightarrow-1-0-f-1-f-1-0\"><pre>\u7b54\u6848\uff1af(-1)=0;\u63d0\u793a\uff1af(x+2)\u4e3a\u5076\u51fd\u6570\\Rightarrow \u5bf9\u79f0\u8f74\u4e3ax=2\uff1bf(2x+1)\u4e3a\u5947\u51fd\u6570\uff0c~\\\\\\Rightarrow f(-2x+1)=-f(2x+1)\\Rightarrow\u5bf9\u79f0\u4e2d\u5fc3\u4e3a(1,0),f(-1)=f(1)=0.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>9.\u5df2\u77e5\u51fd\u6570  f(x)  \u5bf9  \\forall x \\in \\mathbf{R}  \u6ee1\u8db3  f(x+2) \\cdot f(x)=2 f(1) , \u4e14  f(x)&gt;0 . \u82e5  y=f(x-1)  \u7684~~~~\\\\\u56fe\u8c61\u5173\u4e8e  x=1  \u5bf9\u79f0,  f(0)=1 , \u6c42  f(2025).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-2-f-1-x-1-rightarrow-f-1-2-y-f-x-1-x-1-f-x-y-therefore-f-1-f-1-2-therefore-f-x-2-frac-4-f-x-therefore-t-4-therefore-f-2025-f-1-2\"><pre>\u7b54\u6848:2;\u63d0\u793a\uff1a\u5148\u6c42f(1).\u4ee4x=-1,\\Rightarrow f(-1)=2.\u7531y=f(x-1)  \u7684\u56fe\u8c61\u5173\u4e8e  x=1 ~~~~~~\\\\ \u5bf9\u79f0,\u53ef\u5f97f(x)\u5173\u4e8ey\u8f74\u5bf9\u79f0\uff0c\\therefore f(1)=f(-1)=2,\\therefore f(x+2)=\\frac{4}{f(x)},\\therefore\u5468\u671fT=4\uff0c\\\\\\therefore f(2025)=f(1)=2.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>10. (\u591a\u9009) \u5df2\u77e5\u51fd\u6570  f(x) , \u5219\u4e0b\u5217\u547d\u9898\u6b63\u786e\u7684\u4e3a ( ~~~~~~~~)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA. \u82e5  f(1+x)=f(3-x) , \u5219\u51fd\u6570  f(x)  \u7684\u56fe\u8c61\u5173\u4e8e\u76f4\u7ebf  x=2  \u5bf9\u79f0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nB. \u4ee4  g(x)=f(2-x), h(x)=f(2+x) , \u5219\u51fd\u6570  g(x)  \u4e0e  h(x)  \u56fe\u8c61\u5173\u4e8e\u76f4\u7ebf  x=0  \u5bf9\u79f0~~~~~~~~\\\\\nC. \u82e5  f(x)  \u4e3a\u5076\u51fd\u6570, \u4e14  f(x+2)=-f(x) , \u5219\u51fd\u6570  f(x)  \u7684\u56fe\u8c61\u5173\u4e8e\u76f4\u7ebf  x=2  \u5bf9\u79f0~~~~~~~~~~~~\\\\\nD. \u82e5\u51fd\u6570  f(2 x+1)  \u7684\u56fe\u8c61\u5173\u4e8e\u76f4\u7ebf  x=1  \u5bf9\u79f0, \u5219\u51fd\u6570  f(x)  \u7684\u56fe\u8c61\u5173\u4e8e\u76f4\u7ebf  x=2 \u5bf9\u79f0~~\n\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-abc-b-f-x-y-d-f-x-f-2x-1-f-1-x-f-1-x-rightarrow-f-3-2x-f-3-2x-rightarrow-x-3\"><pre>\u7b54\u6848\uff1aABC\uff1b\u63d0\u793a\uff1a\u9009\u9879B\uff0c\u901a\u8fc7f(x)\u56fe\u8c61\u5e73\u79fb\u53ef\u5f97\uff0c\u4e24\u51fd\u6570\u5173\u4e8ey\u8f74\u5bf9\u79f0\uff1b\u9009\u9879D,~\\\\\u4ee4F(x)=f(2x+1),F(1+x)=F(1-x),\\Rightarrow f(3+2x)=f(3-2x),\\Rightarrow\u5bf9\u79f0\u8f74\u4e3a:~~~~\\\\x=3.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>11.\u5df2\u77e5\u51fd\u6570  f(x)  \u7684\u5b9a\u4e49\u57df\u4e3a  \\mathbf{R} , \u4e14  f(x+1)+f(x-1)=2, f(x+2)  \u4e3a\u5076\u51fd\u6570. \u82e5~~~~~~~~~~~\\\\  f(0)=2 ,\u6c42  \\sum_{k=1}^{115} f(k).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-114-f-x-2-rightarrow-f-x-x-2-f-x-2-f-x-2-f-x-2-f-x-2-rightarrow-f-3-x-f-x-1-f-x-1-f-x-1-2-f-3-x-f-x-1-2-rightarrow-1-1-rightarrow-t-4-f-1-1-f-2-0-f-3-1-f-4-2\"><pre>\u7b54\u6848\uff1a114\uff1b\u63d0\u793a\uff1af(x+2)  \u4e3a\u5076\u51fd\u6570\\Rightarrow f(x)\u5173\u4e8ex=2\u5bf9\u79f0\uff0cf(-x+2)=f(x+2)~;\\\\\u7531f(-x+2)=f(x+2)\\Rightarrow f(3-x)=f(x+1),\u4e0e  f(x+1)+f(x-1)=2\u8054\u7acb\u53ef\u5f97~~\\\\f(3-x)+f(x-1)=2\uff0c\\Rightarrow\u5bf9\u79f0\u4e2d\u5fc3\u4e3a(1,1),\\Rightarrow\u5468\u671fT=4\uff0cf(1)=1,f(2)=0,~~~~~\\\\f(3)=1,f(4)=2.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>12.\u5df2\u77e5\u51fd\u6570  f(2 x+1)  \u662f\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u7684\u5947\u51fd\u6570\uff0c\u4e14  f(2 x+1)  \u7684\u4e00\u4e2a\u5468\u671f\u4e3a 2 \uff0c\u5219  (\\quad) ~~~~~\\\\\nA. 1 \u4e3a  f(x)  \u7684\u5468\u671f~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nB.  f(x)  \u7684\u56fe\u8c61\u5173\u4e8e\u70b9  \\left(\\frac{1}{2}, 0\\right)  \u5bf9\u79f0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\nC. f(2023)=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\nD.  f(x)  \u7684\u56fe\u8c61\u5173\u4e8e\u76f4\u7ebf  x=2  \u5bf9\u79f0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-c-f-2-x-1-mathbf-r-rightarrow-f-2x-1-f-2x-1-f-2-x-1-2-rightarrow-f-2-x-2-1-f-2x-1-f-2x-5-f-2x-1-t-4-f-2x-1-f-2x-5-3-0-f-2023-0\"><pre>\u7b54\u6848\uff1aC;\u63d0\u793a\uff1af(2 x+1)  \u662f\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u7684\u5947\u51fd\u6570,\\Rightarrow f(-2x+1)=-f(2x+1),\\Rightarrow \n ~~~~~\\\\f(x)\u7684\u5bf9\u79f0\u4e2d\u5fc3\u4e3a(1,0);f(2 x+1)  \u7684\u4e00\u4e2a\u5468\u671f\u4e3a 2,\u8bbeF(x)=f(2x+1),\u5373F(x+2)=\\\\F(x),\\Rightarrow f(2(x+2)+1)=f(2x+1),\u5373f(2x+5)=f(2x+1),\u5373f(t+5)=f(t+1),\\\\\\Rightarrow f(x)\u5468\u671fT=4.\u7531T=4,\\Rightarrow f(-1)=f(3),\u7531\u5bf9\u79f0\u4e2d\u5fc3\u4e3a(1,0)\uff0c\\Rightarrow f(-1)=-f(3),\\\\\\Rightarrow f(-1)=f(3)=0,f(2023)=f(3)=0\u3002\u9009\u9879D,\u6613\u4e3e\u51fa\u53cd\u4f8b\u4e0d\u6210\u7acb.~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-1111111111111111111111111111111\">\u51fd\u6570\u7684\u5bf9\u79f0\u6027<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u5e38\u89c1\u7ed3\u8bba<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.f(a+x)=f(a-x),f(x)\u5bf9\u79f0\u8f74\u4e3a:x=a;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.f(a+x)=f(b-x),f(x)\u5bf9\u79f0\u8f74\u4e3a:x=\\frac{a+b}{2};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.f(a+x)=-f(a-x),f(x)\u5bf9\u79f0\u4e2d\u5fc3\u4e3a\uff1a(a,0);~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.f(a+x)+f(a-x)=2b,f(x)\u5bf9\u79f0\u4e2d\u5fc3\u4e3a\uff1a(a,b).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-dots\"\/>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5. \u82e5\u51fd\u6570  y=f(x)  \u7684\u5bf9\u79f0\u8f74\u4e3a  x=a, x=b , \u5219\u5176\u5468\u671f\u4e3a  T=2|b-a| .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6.\u82e5\u51fd\u6570  y=f(x)  \u7684\u5bf9\u79f0\u4e2d\u5fc3\u4e3a  (a, 0),(b, 0) , \u5219\u5176\u5468\u671f\u4e3a  T=2|b-a| .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>7. \u82e5\u51fd\u6570  y=f(x)  \u7684\u5bf9\u79f0\u8f74\u4e3a  x=a , \u5bf9\u79f0\u4e2d\u5fc3\u4e3a  (b, 0) , \u5219\u5176\u5468\u671f\u4e3a  T=4|b-a| .~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.\u5df2\u77e5\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u7684\u51fd\u6570  f(x)  \u6ee1\u8db3  f(-x)=-f(x), f(1+x)=f(1-x) , \u5f53  x \\in   [-1,1]  ~~\\\\\u65f6,  f(x)=x^{3}-3 x , \u6c42  f(2023).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-2-x-1-t-4-f-2023-f-1-2\"><pre>\u7b54\u6848\uff1a2\uff1b\u63d0\u793a\uff1a\u5947\u51fd\u6570\uff0c\u5bf9\u79f0\u8f74x=1,\u5468\u671fT=4,f(2023)=f(-1)=2.~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2. \u5df2\u77e5  f(x)  \u662f\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u7684\u5947\u51fd\u6570, \u82e5  f\\left(x+\\frac{3}{2}\\right)  \u4e3a\u5076\u51fd\u6570\u4e14  f(1)   =2 , \u6c42  f(2022)+~~~~~~\\\\f(2023)+f(2024).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-4-x-frac-3-2-therefore-t-6\"><pre>\u7b54\u6848\uff1a4\uff1b\u63d0\u793a\uff1a\u5947\u51fd\u6570\uff0c\u5bf9\u79f0\u8f74x=\\frac{3}{2},\\therefore\u5468\u671fT=6.\u7531\u5bf9\u79f0\u8f74\u5f97f(2)=f(1)=2.~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\u5df2\u77e5  f(x)  \u662f\u5b9a\u4e49\u57df\u4e3a  (-\\infty,+\\infty)  \u7684\u5947\u51fd\u6570, \u6ee1\u8db3  f(1-x)=f(1+   x\uff09 . \u82e5  f(1)=2 , ~~~~~~~~~~\\\\\u6c42 f(1)+f(2)+f(3)+\\cdots+f(50).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-2-t-4-f-1-f-2-f-3-f-4-2-0-2-0-0\"><pre>\u7b54\u6848\uff1a2\uff1b\u63d0\u793a\uff1a\u5468\u671fT=4,f(1)+f(2)+f(3)+f(4)=2+0-2+0=0.~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.\n\u8bbe\u51fd\u6570  f(x)=\\frac{(x+1)^{2}+\\sin x}{x^{2}+1}  \u7684\u6700\u5927\u503c\u4e3a  M , \u6700\u5c0f\u503c\u4e3a  m , \u6c42  M+m.~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-2-f-x-1-frac-2x-sin-x-x-2-1-y-frac-2x-sin-x-x-2-1-0\"><pre>\u7b54\u6848: 2;\u63d0\u793a\uff1a\u51fd\u6570\u53ef\u5316\u7b80\u4e3af(x)=1+\\frac{2x+\\sin x}{x^{2}+1} ,\u5176\u4e2dy=\\frac{2x+\\sin x}{x^{2}+1} \u4e3a\u5947\u51fd\u6570\uff0c~~~~\\\\\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c\u7684\u548c\u4e3a0.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u53d8\u5f0f.\u5df2\u77e5\u51fd\u6570f(x)=ax^3+b\\sin x+4,f(\\lg(\\log _210))=5,\u6c42f(\\lg(\\lg2)).~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-3-f-lg-log-2-10-f-lg-frac-1-lg2-f-lg-lg-2-f-t-5-f-t-3-therefore-f-lg-lg2-3\"><pre>\u7b54\u6848\uff1a3;\u63d0\u793a\uff1af(\\lg(\\log_2 10))=f(\\lg(\\frac{1}{\\lg2}))=f(-\\lg(\\lg 2)),\u82e5\u5df2\u77e5f(t)=5,\u5219~~~~~~~~~~~~~\\\\f(-t)=3\uff0c\\therefore f(\\lg(\\lg2))=3.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5.\u5df2\u77e5\u51fd\u6570  f(x)(x \\in \\mathbf{R})  \u6ee1\u8db3  f(-x)=2-f(x) , \u82e5\u51fd\u6570  y=\\frac{x+1}{x}  \u4e0e  y   =f(x)  \u56fe\u8c61\u7684~~~~~~\\\\\u4ea4\u70b9\u4e3a  \\left(x_{1}, y_{1}\\right),\\left(x_{2}, y_{2}\\right), \\cdots,\\left(x_{m}, y_{m}\\right) , \u6c42  \\sum_{i=1}^{m}\\left(x_{i}+y_{j}\\right)  \u7684\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-m-0-1\"><pre>\u7b54\u6848\uff1am;\u63d0\u793a\uff1a\u4e24\u4e2a\u51fd\u6570\u7684\u5bf9\u79f0\u4e2d\u5fc3\u90fd\u662f(0,1).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6.\u5df2\u77e5\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u7684\u5947\u51fd\u6570  f(x)  \u6ee1\u8db3  f(x+2)=-f(x) , \u5f53  1 \\leqslant x&lt;2 \u65f6,  f(x)=x-2 . ~~~\\\\\u82e5  y=\\frac{1}{6} x-\\frac{1}{3}  \u4e0e  f(x)  \u7684\u56fe\u8c61\u4ea4\u4e8e\u70b9  \\left(x_{1}, y_{1}\\right),\\left(x_{2}, y_{2}\\right), \\cdots,\\left(x_{n}, y_{n}\\right)\\left(n \\in \\mathbf{N}^{*}\\right) ,\u6c42~~~~~~~~~~~~~~\\\\  \\sum_{i=1}^{n}\\left(x_{i}+y_{i}\\right).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-14-2-0\"><pre>\u7b54\u6848\uff1a14;\u63d0\u793a\uff1a\u4e24\u4e2a\u51fd\u6570\u7684\u5bf9\u79f0\u4e2d\u5fc3\u90fd\u662f(2,0).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-111111111111111111111111111111111111111\">\u5bfc\u51fd\u6570\u4e0e\u539f\u51fd\u6570\u7684\u5bf9\u79f0\u6027<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u5e38\u89c1\u7ed3\u8bba<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(1)f(x)\u4e3a\u5947\u51fd\u6570\uff0c\u5219f^{\\prime}(x)\u4e3a\u5076\u51fd\u6570;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(2)f(x)\u5bf9\u79f0\u4e2d\u5fc3\u4e3a(a,b)\uff0c\u5219f^{\\prime}(x)\u5bf9\u79f0\u8f74\u4e3ax=a;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(3)f(x)\u4e3a\u5076\u51fd\u6570\uff0c\u5219f^{\\prime}(x)\u4e3a\u5947\u51fd\u6570;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(4)f(x)\u5bf9\u79f0\u8f74\u4e3ax=a,f^{\\prime}(x)\u5bf9\u79f0\u4e2d\u5fc3\u4e3a(a,0);~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(5)f(x)\u5468\u671f\u4e3aT,f^{\\prime}(x)\u7684\u5468\u671f\u4e3aT.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.(\u591a\u9009)\u5df2\u77e5\u51fd\u6570  f(x)(x \\in \\mathbf{R})  \u662f\u5947\u51fd\u6570,  f(x+2)=f(-x)  \u4e14  f(1)=2, f^{\\prime}(x) \u662f  f(x)  \u7684~~~\\\\\u5bfc\u51fd\u6570\uff0c\u5219  (\\quad) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\A.  f(2025)=2 ~~\nB.  f^{\\prime}(x)  \u7684\u5468\u671f\u662f 4~~\nC.  f^{\\prime}(x)  \u662f\u5076\u51fd\u6570~~\nD.  f^{\\prime}(1)=1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-3-f-lg-log-2-10-f-lg-frac-1-lg2-f-lg-lg-2-f-t-5-f-t-3-therefore-f-lg-lg2-31\"><pre>\u7b54\u6848\uff1aABC;\u63d0\u793a\uff1af(x+2)=f(-x)\\Rightarrow f(x)\u5bf9\u79f0\u8f74\u4e3ax=1,\\Rightarrow f^{\\prime}(x) \u5bf9\u79f0\u4e2d\u5fc3~~~~~~~~~~~\\\\\u4e3a(1,0),\u53c8f(x)\u4e3a\u5947\uff0c\\Rightarrow f(x)\u5468\u671fT=4,f^{\\prime}(x) \u5468\u671f\u4e5f\u662f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.\u5df2\u77e5\u51fd\u6570  f(x)  \u53ca\u5176\u5bfc\u51fd\u6570  f^{\\prime}(x)  \u7684\u5b9a\u4e49\u57df\u5747\u4e3a  \\mathbf{R} , \u8bb0  g(x)=f^{\\prime}(x) ,\u82e5  f(x+2)  \u4e3a\u5076\u51fd\u6570\uff0c\\\\  g(x)  \u4e3a\u5947\u51fd\u6570\uff0c\u5219  (\\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\\\\nA.  f(x)=f(4-x) ~~\nB.  g(x)=-g(4-x) ~~\nC.  f(x)=-f(x+4) ~~\nD.  g(x)=g(x+4) ~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-3-f-lg-log-2-10-f-lg-frac-1-lg2-f-lg-lg-2-f-t-5-f-t-3-therefore-f-lg-lg2-311\"><pre>\u7b54\u6848\uff1aABD;\u63d0\u793a\uff1af(x+2)\u4e3a\u5076\uff0cg(x)\u5bf9\u79f0\u4e2d\u5fc3\u4e3a(2,0),\u53c8g(x)\u4e3a\u5947\uff0c\\therefore g(x)\u5468\u671f~~~~\\\\T=4,\\therefore BD\u6b63\u786e.f(x+2)  \u4e3a\u5076\u51fd\u6570\uff0c\\therefore f(x)\u5bf9\u79f0\u8f74\u4e3ax=2,A\u6b63\u786e.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\u5df2\u77e5\u51fd\u6570  f(x)  \u662f\u5b9a\u4e49\u5728  \\mathbf{R}  \u4e0a\u7684\u53ef\u5bfc\u51fd\u6570, \u5176\u5bfc\u51fd\u6570\u4e3a  g(x), f(x+2)  \u548c  g(x+1)  \u90fd\u662f\u5947\\\\\u51fd\u6570,  f(1)=1 , \u5219\u4e0b\u5217\u8bf4\u6cd5\u6b63\u786e\u7684\u662f (~~ )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  g(x)  \u5173\u4e8e\u70b9  (1,0)  \u5bf9\u79f0~~\nB.  f(x)+f(-x)=0 ~~\nC.  g(2025)=1 ~~\nD.  \\sum_{k=0}^{2024} f(k)=0 ~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-da1\"><pre>\u7b54\u6848\uff1aABD;\u63d0\u793a\uff1a g(x)=f^{\\prime}(x)\uff0cf(x+2)\u4e3a\u5947\u51fd\u6570\uff0c\\Rightarrow f(x)\u5bf9\u79f0\u4e2d\u5fc3\u4e3a(2,0)\u4e14g(x)\\\\\u5bf9\u79f0\u8f74\u4e3ax=2;g(x+1)\u4e3a\u5947\u51fd\u6570\uff0c\\Rightarrow g(x)\u5bf9\u79f0\u4e2d\u5fc3\u4e3a(1,0)\u4e14f(x)\u5bf9\u79f0\u8f74\u4e3ax=1.~~~~~<\/pre><\/div>\n<\/details>\n\n\n\n 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