名校
1 . 已知定义在
上的函数
满足以下三个条件:
①对任意实数
,都有
;
②
;
③
在区间
上为增函数.
(1)判断函数
的奇偶性,并加以证明;
(2)求证:
;
(3)解不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
①对任意实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acf84c184be32752d1c14e6f23fecda8.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6855784817151468771f29c0fc38fc9.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304226ca50149b49702928e44d565964.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4cff510b81f7160ec53b7ef179f114.png)
(3)解不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5be1d8c6384d7fabddb693b2b7fcdf4a.png)
您最近一年使用:0次
2019-12-01更新
|
929次组卷
|
3卷引用:上海市复旦大学附属中学2022届高三上学期9月月考数学试题
名校
2 . 已知函数
其反函数为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
(1)求证:对任意
都有
,对任意
都有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c70f029102bd0b5e762717c3889671fb.png)
(2)令
,讨论
的定义域并判断其单调性(无需证明).
(3)当
时,求函数
的值域;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bb2fb6043949ffd4a0fc14967e23c90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
(1)求证:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dcbca3478eae63853d2aab5332e2e56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0b8e9b3f07d91da4d256d18df240fe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edae93ec9de65d7e8afd2a53063c8ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c70f029102bd0b5e762717c3889671fb.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57a76b586e289841016c49819b99559f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd0f5e152398772be9ec9555664a6407.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ee2ec7a69d7d1f401e04afd231f6515.png)
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名校
3 . 已知定义在
上的函数
满足:对任意
都有
.
(1)求证:函数
是奇函数;
(2)如果当
时,有
,试判断
在
上的单调性,并用定义证明你的判断;
(3)在(2)的条件下,若
对满足不等式
的任意
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/455ba3d3e46977fcbe5b71f8bb9df4be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c75a15990fdcf1de0a9ac9f475e3c92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49074b2fc18e7edb1b3b6b4e6f9737c9.png)
(1)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)如果当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e01d07f3a82196cabb98a2ab98686eb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a71baf6217604517fd98fa97d0f55b43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/455ba3d3e46977fcbe5b71f8bb9df4be.png)
(3)在(2)的条件下,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ab785be07d4b90f42c992e4d7b2f8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/159ece6e1d45537def7b40aef5083146.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
2019-10-26更新
|
676次组卷
|
3卷引用:江西省抚州市南城县第二中学2019-2020学年高一上学期第一次月考数学试题
名校
4 . 对于定义域为D的函数y=f(x),如果存在区间[m,n]
D,同时满足:
①f(x)在[m,n]内是单调函数;
②当定义域是[m,n]时,f(x)的值域也是[m,n].则称[m,n]是该函数的“和谐区间”.
(1)证明:[0,1]是函数y=f(x)=x2的一个“和谐区间”.
(2)求证:函数
不存在“和谐区间”.
(3)已知:函数
(a∈R,a≠0)有“和谐区间”[m,n],当a变化时,求出n﹣m的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/637904facd16726fbfccb679e901e68a.png)
①f(x)在[m,n]内是单调函数;
②当定义域是[m,n]时,f(x)的值域也是[m,n].则称[m,n]是该函数的“和谐区间”.
(1)证明:[0,1]是函数y=f(x)=x2的一个“和谐区间”.
(2)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d3b0573a4ee2c68c86feda380291faf.png)
(3)已知:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a087c10b183ee28bc5fe1faa3289074.png)
您最近一年使用:0次
2016-12-04更新
|
1243次组卷
|
8卷引用:上海市上海实验学校2019-2020学年高三上学期9月第一次月考数学试题
5 . 给出以下两个数学运算(符号)定义:
①若函数
,则
,其中
称为函数
的
次迭代.如:
.
②对于正整数
,若
被
除得的余数为
,则称
同余于
,记为
.如:
.
(1)若函数
,求
;
(2)设
是一个给定的正整数,函数
记集合
.
①证明:当
时,
;
②求
并猜想
.
①若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac36cc9de2d52fa81b310df3c137559f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bbc5dcd0b5d4e94cb92e52ca31f0cfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b956f95d6cdcded732751d6d74c14cab.png)
②对于正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0ace40e2e209924905e48bf00df631f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8b539bc9386988afc25da70e13ae899.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/706057125d5d481d23b0319e10e2d936.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e08e564833f8450a876460a6db43dad1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e91f1470cecf7c4da36644e5244775bf.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b44a1741d645756e39740a0818412e75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/208bd4b35f6be79500cdf5d8e433e449.png)
①证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b94f6525788e512dbc8121c49b46bc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/502416314c8c26f8442e639ea6a5db13.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00442d96d695db2c58bf1fb7165fca94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
您最近一年使用:0次
名校
6 . 如果对于函数
的定义域内任意的
,都有
成立,那么就称函数
是定义域上的“平缓函数”.
(1)判断函数
是否是“平缓函数”;
(2)若函数
是闭区间
上的“平缓函数”,且
,证明:对于任意的
,都有
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c2f1ca03ade14de6711c85de8fc5df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ea19565e4feac073e898ab188fc3f5.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aeb3ca8cbc4facb2467b1a618f33794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83a6c0fddb9074dfc96be03b4aa24d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14e9387190a323961884c302798c9e4e.png)
您最近一年使用:0次
7 . 已知函数
.
(1)若
,求
的取值范围;
(2)若
有两个不相等的实根
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
①求
的取值范围;
②证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bfe72da09a3c70855e5481e5c8eaf80.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99a1b101171cf52faf2e3e53d6725aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a18ca67c2770b98f36dbfd802595a95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
②证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dd25b0aac17082018e89b67803cd6a1.png)
您最近一年使用:0次
8 . 设集合、
为正整数集
的两个子集,
、
至少各有两个元素.对于给定的集合
,若存在满足如下条件的集合
:
①对于任意,若
,都有
;②对于任意
,若
,则
.则称集合
为集合
的“
集”.
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55a4c458e37238547c09a481eb0ca295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
(2)若三元集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275bd8ce17fcc4a786510b008414ab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275bd8ce17fcc4a786510b008414ab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21edab576db2b4f56237ad687957a913.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/380cc5e32580ddf29206ebb596336151.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea5867fde790c54e6a931c5d1d22b049.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
您最近一年使用:0次
解题方法
9 . 对于数集
,
,定义向量集
,若对任意
,存在
使得
,则称X是“对称的”.
(1)判断以下三个数集
、
、
是否是“对称的”(不需要说明理由);
(2)若
,且
是“对称的”,求
的值;
(3)若“对称的”数集
,
满足:
,
,
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c61b6f4ad8f11fa9c6e5268b5368df3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d80db4c6ae227b62067e092f740e7a41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eec1c65f144bd63ed516e001e57852de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f923fcc615e579b8dda937faa9fa40c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01243e3fb9bd7a7711a593f5395b06cd.png)
(1)判断以下三个数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21c6fed9c3cf2c00ba1823c3f0a05615.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ee021c7c1a5df78501eaca655726212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/939f7dc30e48606f0aafd5ab6d9a93b5.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/752455799e49f846e2601304fec5d3b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41130c870a38d91008b7019ae296feca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(3)若“对称的”数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c61b6f4ad8f11fa9c6e5268b5368df3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4049b329e8cf711663e050e0dc9cdea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/007defcff0a2cfbbb6fade9a3ab53bcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/346549f9adda7eb363f16d355ae68b85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7eba583e37243f3ba166bd1c11e58498.png)
您最近一年使用:0次
名校
解题方法
10 . 对于定义域在
上的函数
,定义
.设区间
,对于区间
上的任意给定的两个自变量的值
、
,当
时,总有
,则称
是
的“
函数”.
(1)判断函数
是否存在“
函数”,请说明理由;
(2)若非常值函数
是奇函数,求证:
存在“
函数”的充要条件是存在常数
,使得
;
(3)若函数
与函数
的定义域都为
,且均存在“
函数”,求实数
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d25597c0f369019a0901849bc12da1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1cb71b8c83c4f5a3146e3871b6308d4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.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/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c61c8d37c767ba727cc7f5f7e00a7d96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d6f99885e464b84f1dc2b897070cbdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)若非常值函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0d314b6f3729e70a0d0c60414aec69c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2c9418985f008bb9ab6482930f187dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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(3)若函数
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6卷引用:上海市东华大学附属奉贤致远中学2023-2024学年高一上学期12月教学评估数学试题
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