名校
1 . 已知函数
,
为函数
的反函数
(1)讨论
在
上的单调性,并用定义证明;
(2)设
,求证:
有且仅有一个零点
,且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e4ccec118032fd96e0713b04c3a27a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12be206d66e65eb92ef08bad8cd8f71d.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae4a2b3998705e51dbade9ada0873b2b.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041f581f277a2de1ef41c354b6e6991e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c49ca8562b98657ca9c499093f7233.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7d126a2ae5babaf18b9082a975cdc52.png)
您最近一年使用:0次
2 . 已知函数
,
,且满足
.
(1)求实数a的取值范围;
(2)求证函数
存在唯一零点;
(3)设
,证明
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fb0c7b952731190aea730a9fb18a603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66692ec49a458f9e48c7315d03dfc37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a02872c8c4d0f941ad55b2f88fa58ea.png)
(1)求实数a的取值范围;
(2)求证函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c419949314258c61e4436e16477fa42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67a51414243ca45bcca00d14a9865f93.png)
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3 . 对任意正整数n,记集合
,
.
,
,若对任意
都有
,则记
.
(1)写出集合
和
;
(2)证明:对任意
,存在
,使得
;
(3)设集合
.求证:
中的元素个数是完全平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a39352d44787ecda055946f530893f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5359f5f086b89cc656cdd4f79a3b7baa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5df0913f90131b298c8f6f57437f69b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c1352dca7dc3caf67c1cb937d52795.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d35755ad07f05e7bfe00176d6334389f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75a28908216e6879a09b372d957be1e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90df641ab645927ee577e79faf18dcdd.png)
(1)写出集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a71fc9c0068109dad1382354570665.png)
(2)证明:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e840ba7606959ccea36793f3ef0775d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b576952835af1f3492f0f3e6d00093e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90df641ab645927ee577e79faf18dcdd.png)
(3)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/112d102cbe7ce2bebe0c76e87e89a00c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
您最近一年使用:0次
2023-11-15更新
|
161次组卷
|
4卷引用:北京市朝阳区2022届高三上学期期末统一检测数学试题
解题方法
4 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ad292a5e3f68651844e4207b9b594bf.png)
(1)直接写出函数
的零点和不等式
的解集;
(2)直接写出函数
的定义域和值域;
(3)求证:函数
的图象关于点
中心对称;
(4)用单调性定义证明:函数
在区间
上是减函数;
(5)设
,直接写出它的反函数
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ad292a5e3f68651844e4207b9b594bf.png)
(1)直接写出函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73a98c1b3504e09bfbe0db849b0d24.png)
(2)直接写出函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b334e2eaa7e8fb79cef8208b56ee4f5.png)
(4)用单调性定义证明:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e10140ab3cdc13d710a65b2287c892b.png)
(5)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e313b39064db7bfb103e6215440b19e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc58675aca9c02251a17d4fca67ea5dd.png)
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名校
5 . 定义:有限非空数集
的所有元素的“乘积”称为数集
的“积数”,例如:集合
,其“积数”
.
(1)若有限数集
,求证:集合
的所有非空子集的“积数”之和
满足
;
(2)根据(1)的结论,对于有限非空数集
(
),记集合A的所有非空子集的“积数”之和
,试写出
的表达式,并利用“数学归纳法”给予证明;
(3)若有限集
,
①试求由
中所有奇数个元素构成的非空子集的“积数”之和
奇数;
②试求由
中所有偶数个元素构成的非空子集的“积数”之和
偶数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb9ad1e34877b0db02d0219332b0f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb9ad1e34877b0db02d0219332b0f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/635cc4bb9a743b88c98fffad8ba1af00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc5787e5d2863aa157213424a4803245.png)
(1)若有限数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d020cd453031ae9eede7961ec78f21a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2319b6a5373bc8eb13772b8e6d047779.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b64379aceaa2d008a48356937130c9e.png)
(2)根据(1)的结论,对于有限非空数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ea7fcdb5423c1c8c032a3efcf245682.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/576ea0f23e66276d14e99a90c149c0dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(3)若有限集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f994206101b7f04f92c5d4e2dcae7b8d.png)
①试求由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb9ad1e34877b0db02d0219332b0f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
②试求由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb9ad1e34877b0db02d0219332b0f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
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名校
6 . 我们知道一次函数、二次函数的图像都是连续不断的曲线,事实上,多项式函数的图像都是如此.
(1)设
,且
,若还有
,求证:
;
(2)设一个多项式函数有奇次项
(
),求证:总能通过只调整
的系数,使得调整后的多项式一定有零点;
(3)现有未知数为
的多项式方程
(其中实数
待定),甲、乙两人进行一个游戏:由甲开始交替确定
中的一个数(每次只能去确定剩余还未定的数),当甲确定最后一个数后,若方程由实数解,则乙胜,反之甲胜,问:乙有必胜的策略吗?若有,请给出策略并证明,若无,请说明理由.
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dceea9a267bf6a1a79a2b1be84dc8b9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/292a756873e88b6e90ddc8d9711cc6da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6a02e169b7678c8b3741cb187299c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acf237f6c2170d7c7fb27acbafd16f64.png)
(2)设一个多项式函数有奇次项
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db98876d40d5afd3ba01c668e96e9d0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e10f2f74e201f77f853e9ed9078615c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db98876d40d5afd3ba01c668e96e9d0e.png)
(3)现有未知数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69d41e428667bede26795a0401ddcd77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52812f95d26eec5dcd489b076cd35718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1188fbc777615a17789b1fb54fcb7e34.png)
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名校
7 . 已知函数
,函数
是函数
的反函数.
求函数
的解析式,并写出定义域
;
设
,判断并证明函数
在区间
上的单调性:
若
中的函数
在区间
内的图像是不间断的光滑曲线,求证:函数
在区间
内必有唯一的零点(假设为
),且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/685b979275f63408d20543770df4f2ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe5853a3e36e55ccf04a974c6df2811.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bf6c84731e5e1bd335ecfc2d36c3d81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f53190d6ead827a6338b9de847aeaf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9abbcaa32b0525269d0cb445cabaa870.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f786a5701dc1a8a015e8843c3360151b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f62295c36d2e2174908c2bec0eb5b30f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f53190d6ead827a6338b9de847aeaf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f786a5701dc1a8a015e8843c3360151b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f786a5701dc1a8a015e8843c3360151b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a2a5e336b6bcba6354fd366c892dd06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60440d5dde56b026d8568075463a988a.png)
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8 . 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:浙江省宁波市效实中学2019-2020学年高一上学期期中数学(理)试题
9 . 对于定义域为D的函数
,如果存在区间
,同时满足:①
在
内是单调函数;②当定义域是
时,
的值域也是
.则称
是该函数的“和谐区间”.
(1)证明:
是函数
=
的一个“和谐区间”.
(2)求证:函数
不存在“和谐区间”.
(3)已知:函数
(
R,
)有“和谐区间”
,当
变化时,求出
的最大值.
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/85788af6b4a64af49a2488b14790cbc4.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/0c9cc65ece4c41f7932a390bb4a491c1.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/86162c78c4b144bc89a2c748a040b308.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/5bfa40ca62b848a4b0515b76807276ec.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/5bfa40ca62b848a4b0515b76807276ec.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/86162c78c4b144bc89a2c748a040b308.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/5bfa40ca62b848a4b0515b76807276ec.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/5bfa40ca62b848a4b0515b76807276ec.png)
(1)证明:
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/f237254e258b4ec281e12610b5d7e5ab.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/85788af6b4a64af49a2488b14790cbc4.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/63c0d3e3823644e5bbe2efe41ffe1590.png)
(2)求证:函数
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/860a31536a6b4cbba385cb94a18d53cf.png)
(3)已知:函数
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/903023ddba954478acf160b661848db1.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/80ca0bb0234f4b819f857dd8814e6fa2.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/5b6cb3b1916a44acbeee023fcd25fee7.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/5bfa40ca62b848a4b0515b76807276ec.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/931f1a47f3fd41e6bd63d40181e59177.png)
![](https://img.xkw.com/dksih/QBM/2016/11/25/1573182759813120/1573182766161920/STEM/036270e93bff4c29880b98c7701723d3.png)
您最近一年使用:0次
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解题方法
10 . 函数
的定义域为D,若存在正实数k,对任意的
,总有
,则称函数
具有性质
.
(1)判断下列函数是否具有性质
,并说明理由;
①
;
②
;
(2)已知
为二次函数,若存在正实数k,使得函数
具有性质
.求证:
是偶函数;
(3)已知
,k为给定的正实数,若函数
具有性质
.求a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a380348dd1544f954255976659a84a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daac43c7675fa411b35028e09b0bad90.png)
(1)判断下列函数是否具有性质
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2387880727d458702651d699e76d7d76.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5bc39061e1fb75d8ab1fd5c3765a514.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4306fb6d5419322b4b7b9140e06e43a0.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daac43c7675fa411b35028e09b0bad90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d88a63bbc4eadab1cbce4dbfaf8d411.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daac43c7675fa411b35028e09b0bad90.png)
您最近一年使用:0次
2023-11-24更新
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237次组卷
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2卷引用:湖南省长沙市周南中学2022-2023学年高二上学期暑假学习评价检测数学试题