1 . 设集合
.定义:和集合
,积集合
,分别用
表示集合
中元素的个数.
(1)若
,求集合
;
(2)若
,求
的所有可能的值组成的集合;
(3)若
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59889190d638327c0b0b0cc162d41221.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f706563c8dc02e70264ce4211a89c709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eaec9915cb9f6c01cb87a6c7b0d121b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2c376afabc2928dd4299dca55d5bfc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a9e6ad1166c7625e63b80e75b2fb1d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d2734b136f9961df15bb51c31e29e28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8f054b30b493c4ce1deecff3468c08.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d197338ab810f6c9d31a2b67e5f352ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ef01842529a2a0028f17f35b6281e1a.png)
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2 . 已知函数
和
的定义域分别为
和
,若对任意
,恰好存在n个不同的实数
,
,…,
,使得
(其中
,2,…,n,
),则称
为
的“n重覆盖函数”.
(1)判断
(
)是否为
(
)的“n重覆盖函数”,如果是,求出n的值;如果不是,说明理由;
(2)若
为
的“2重覆盖函数”,求实数a的取值范围;
(3)函数
表示不超过x的最大整数,如
,
,
,若
,
为
,
的“2024重覆盖函数”,求正实数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.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/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc40290feeafceca34cbdab068dcd769.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe44a5aed663a9b61ef7355b38c77d0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c45176df950dfe48b8ca7eac08ee349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/986cf0c6c92f8a0e0370b50d5606d483.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/365114c53aa12abda1004c8e4cb4ca0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c1b4a895643b6b3cfddbe5d4a1157b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8160ade472b89421f8009fd0cd3926a3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f246e5b05b68bb9fdeb12a319aa7136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa88c20e58953bba4ed04d3ce419df95.png)
(3)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fe1e778c9e668594c42b77459328c63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3420606c96b68fb884c839923fd20a25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf813e9500eebd474511b865b876ea4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa0c6435833958c7987934c3367d48b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50af11c345056215054f7cfe679939da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a5877b36b0def7389b8fb66e8491644.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac4cbc7b067862a3d9c6789b392fc068.png)
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3 . 已知函数
为常数).函数
定义如下:对每个给定的实数
.
(1)若
,求
在
上的最大值;
(2)若
且
,求函数
在区间
上的单调增区间的长度之和.(闭区间
的长度定义为
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/095878335d1f0c342afc4a2a300edfe6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b426608a06477f57cb994f4d00e4465d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68f842876e8f512903598574fa62e86e.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20e1c681b27df538bd4742f6cd8298ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b426608a06477f57cb994f4d00e4465d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7242b2ab643f9470da77e29d043b893.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/575002003f89bc8309a2c7f206761794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfa21416f901e005780f383fb029e86f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b426608a06477f57cb994f4d00e4465d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede949a2fe7bf9f1f98f5457d408b41d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57b85a97933a1d984f6e484b4021c800.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64da75a02173c2a5eb40f4c68d0f4f36.png)
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4 . 已知函数
的定义域均为
,且
,
,若
的图象关于直线
对称,则以下说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c0ed188d083966baaae94e6b86064f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/614af486f69e73f4fb0e23f0e686fc9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96e157b2495c34c099ec2c22a67ee898.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
A.![]() | B.![]() |
C.![]() ![]() | D.若![]() ![]() ![]() |
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5 . 已知函数
的定义域为
为奇函数,
为偶函数,当
时,
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2becb8c04d632ce418ee8a467967f1b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2d0e9cd42a043be4c6e552c6cf4cbdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0810d505a46fc0c3a11eee665469646e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be5ff6072bb799712801e9375d433783.png)
A.![]() |
B.![]() |
C.![]() ![]() |
D.方程![]() |
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6 . 已知函数
,若函数
有三个不同的零点
,且
,则
的取值范围是_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4bed8b04ab17cccbbc4ec882b08f5f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b8ec9d4206ea66a02de5c4a1e1e911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52079d948ae0e1f1d112b3ac76142f1a.png)
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7 . 已知函数
,
,其中
,
.
(1)若
,求函数
在
上的值域;
(2)若函数
恰好有三个零点
,且
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96a72eeeef8313692965bf5ddae91407.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a56807f295dc1a2c56a2cb2fb70dc42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cf2f46bc120e5b6f8317b02370bffbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b108ab31cc093f03cf48ad65429889e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d188ec2580e273ce87e51653a2177ee.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa662f0273f0921c1fa4727f632395.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29ba2c12fa14a6f63de6b628b495393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98f65ae14389e6904956bd88df4450e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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8 . 已知函数
,则函数
的零点有______ 个;关于
的方程
的实根个数构成的集合为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33cf0cb49254dfcfd1729c329c8f8fbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b670e3986cd86a6363100498d67c5430.png)
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9 . 已知定义在R上的函数
,满足对
,有
,则称
为“好函数”.下列说法中正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afd5bcbbf845d7595b1636d751e718d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/458b8b961b627061cd1ac4ca07ab9564.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
A.若![]() ![]() |
B.若![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() ![]() ![]() |
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10 . 已知函数
,
,其中
,
.
(1)求函数
在
上的最小值;
(2)若函数
恰好存在三个零点
、
、
,且
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96a72eeeef8313692965bf5ddae91407.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a56807f295dc1a2c56a2cb2fb70dc42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e10e1c43b86a8cd4360ca9b57232164.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d188ec2580e273ce87e51653a2177ee.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa662f0273f0921c1fa4727f632395.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/72f15a6bfa65d50ee3fee7d4eff24b89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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