1 . 若定义域为
的奇函数
满足
,则
在
上的零点个数至少为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dac47c1b6230edf33b5a1c76b75025de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b61bb7cb94b4d06f0090df1e365667.png)
A.5 | B.6 | C.7 | D.8 |
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解题方法
2 . 已知函数
和
的定义域分别为
和
,若对任意
,恰好存在
个不同的实数
,
,使得
(其中
),则称
为
的“
重覆盖函数”.
(1)判断
是否为
的“
重覆盖函数”,如果是,求出
的值;如果不是,请说明理由;
(2)若
为
的“3重覆盖函数”,求实数
的取值范围;
(3)若
为
的“2024重覆盖函数”,求正实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6795cae2df43a722e1355e9562d93c09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c296e45b84cf67a98939aa7334e7d478.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3eddf991be37d25d033f78bd3511809.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73ecbf34f484b62628e762bb12f09ffa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95d3de6d381d271e1098162aa98a654e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e892e3c7dde74f267608ab5b09d22eb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ccf84b235ba036b51ca8e53ae6507a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b948f98a521c5d18fc3dfa60bbbe8a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92e94a2262e0c51949ee65333435048a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62797a22eb499fc7317f67b1e178b568.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41fea50313112bf7f6651b442346acf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb2687a5b4ded9fa85f5ae826ebb9e9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
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3 . 下列说法正确的是( )
A.已知方程![]() ![]() ![]() |
B.函数![]() ![]() |
C.函数![]() |
D.用二分法求函数![]() ![]() ![]() ![]() |
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解题方法
4 . 设定义在
上的函数
满足
,
,且
时,
,则方程
在区间
上所有实数根的和为_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e298fe246eef819dd9b1edabe3bb9cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01bea8bf593f594c51fc7cc547482bee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd684b496b0d74004440ec0d72cea763.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/151fecd2b966c4ab88aba676bf0590c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e702628127ed874fb0960aa47151b30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e79dee388c7c7f23b078320707ab0f04.png)
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5 . 已知函数
,下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5233d925bf28a4fd8f4d70858b6396e.png)
A.![]() |
B.方程![]() |
C.当![]() ![]() |
D.曲线![]() ![]() |
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解题方法
6 . 定义在
上的奇函数
满足
,当
时,
,则函数
的零点的个数为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86d78dec1c1e00ec02d7bdaf76ef8901.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f5c837522a811402efb9762210c5362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29e44284cb19805a584880a686ac3df9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e6a5e22ade871e35970d0eb59a3352.png)
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名校
解题方法
7 . 若函数
满足
且
,则称函数
为“M函数”.
(1)试判断
是否为“M函数”,并说明理由;
(2)函数
为“M函数”,其在
的图象落在直线
上,在函数
图象上任取一点P,对于定点
,求线段AP的最小值;
(3)函数
为“M函数”,且当
时,
,求
的解析式;若当
,关于x的方程
(a为常数)有解,记该方程所有解的和为S,求S.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ebe7bbef9a9ac82bdcd7a4d619062ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0caa479f637ac7494110301dd19df178.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
(1)试判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a00d098f39c3266b26d52782726e475f.png)
(2)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/379c69aa8be40d6a181902946d6f85bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/587a19adac2eaac76b64d2c400ae06ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c77d6b8fb57e9c98dd483b11a6100876.png)
(3)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090b1afae79da26a29bc81dd7f5a179b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3162d2c7b650bba3e401ffbb1e13bb45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04e2da78cacf1082b7ac6aea946f7208.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0da3023b0765cfb1b268e29e1d01de0e.png)
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解题方法
8 . 已知函数
,其中
.
的零点;
(2)讨论关于x的方程
的解的个数;
(3)若方程
有四个不同的根
,
,
,
,直接写出这四个根的和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d38ec862e64efb2e3001675db12be9d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)讨论关于x的方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb291880ef86317d079c0e0b349403e5.png)
(3)若方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb291880ef86317d079c0e0b349403e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/365b38a7689a8eede6820cd6f1fe952b.png)
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2024高三·全国·专题练习
解题方法
9 . 已知函数
,若
,且
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5304df517a9f107262ef8e727f88af98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3604274ad6707a906eba371a9e884144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2714f1eb0e74a1a31fa2c35275d368e1.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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解题方法
10 . 材料一:我们可以发现这样一个现象:随机生成的一元多项式,在复数集中最终都可以分解成一次因式的乘积,且一次因式的个数(包括重复因式)就是被分解的多项式的次数.事实上,数学中有如下定理:
代数基本定理:任何一元
次复系数多项式方程
至少有一个复数根.
材料二:由代数基本定理可以得到:任何一元
次复系数多项式
在复数集中可以分解为
个一次因式的乘积.进而,一元
次多项式方程有
个复数根(重根按重数计).
下面我们从代数基本定理出发,看看一元多项式方程的根与系数之间的关系.
设实系数一元二次方程![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15f23a2c5600a7c4dc3658055ef091e3.png)
在复数集
内的根为
,容易得到![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a877d84778ecc1d7999803eba414102.png)
设实系数一一元三次方程
①
在复数集
内的根为
,可以得到,方程①可变形为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c5f02ca9521a8d68480025eaf893e95.png)
展开得:
②
比较①②可以得到根与系数之间的关系:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67f2ae75fee77fee30aa151798182849.png)
阅读以上材料,利用材料中的方法及学过的知识解决下列问题:
(1)对于方程
在复数集
内的根为
,求
的值;
(2)如果实系数一元四次方程
在复数集
内的根为
,试找到根与系数之间的关系;
(3)已知函数
,对于方程
在复数集
内的根为
,当
时,求
的最大值.
代数基本定理:任何一元
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6368fec0c2c25db7c29b014d60270e97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
材料二:由代数基本定理可以得到:任何一元
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6368fec0c2c25db7c29b014d60270e97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
下面我们从代数基本定理出发,看看一元多项式方程的根与系数之间的关系.
设实系数一元二次方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15f23a2c5600a7c4dc3658055ef091e3.png)
在复数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b222b256b37f83fa24a3a4b6527f58d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a877d84778ecc1d7999803eba414102.png)
设实系数一一元三次方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1667cf29d96c1cd87841816a6d99c8c3.png)
在复数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67f266b8ffa43e3ed5302060fbee8dc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c5f02ca9521a8d68480025eaf893e95.png)
展开得:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e35119b570f422658c3c4df87db6a62d.png)
比较①②可以得到根与系数之间的关系:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67f2ae75fee77fee30aa151798182849.png)
阅读以上材料,利用材料中的方法及学过的知识解决下列问题:
(1)对于方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18b886eb137027de95b12a407da3002e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67f266b8ffa43e3ed5302060fbee8dc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/019980a9716b372a9b8e119847be1510.png)
(2)如果实系数一元四次方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50008c76ed92c7d212ccce2f7c786ad8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a00d3e1c1ae2917f675860ccbb4959a2.png)
(3)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e212fdca109324877fcb9c280306368.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19d3d7b48ea2c4a0c74ebfb263c4eeb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67f266b8ffa43e3ed5302060fbee8dc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/716baa5af4df2ef1e24f1bdb496073cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5bc4081a1bbf3e7b0a1c856975a0b9e.png)
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