解题方法
1 . 在数学中,不给出具体解析式,只给出函数满足的特殊条件或特征的函数称为“抽象函数”.我们需要研究抽象函数的定义域、单调性、奇偶性等性质.对于抽象函数
,当
时,
,且满足:
,均有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367936b458618efb6b2eadc843e5d6ba.png)
(1)证明:
在
上单调递增;
(2)若函数
满足上述函数的特征,求实数
的取值范围;
(3)若
,求证:对任意
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73a98c1b3504e09bfbe0db849b0d24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6571b33b56c6cd88f2f6e091031bcf40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367936b458618efb6b2eadc843e5d6ba.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639c5f8b7a1a268c904d04356f0d1b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be9b79f42bbf0de1851607050c3e8d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/219598f1289ddb370d632ea141731d52.png)
您最近一年使用:0次
名校
解题方法
2 . 函数
(
且
)是定义在R上的奇函数.
(1)求a的值,并判断
的单调性,并证明;
(2)若存在
,使得
成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d8b85ce9b066e972f9e94f1b9932b06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
(1)求a的值,并判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b008beb08962361a5e035b2989c4d5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef92f9154725b84be418f9e73ca1d33f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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2024-01-30更新
|
462次组卷
|
3卷引用:广东省广州二中2023-2024学年高一上学期期末数学试题
解题方法
3 . 平原上两根电线杆间的电线有相似的曲线形态,这些曲线在数学上称为悬链线.悬链线在工程上有广泛的应用.在恰当的坐标系中,这类曲线的函数表达式可以为
,其中a、b为非零实数
(1)利用单调性定义证明:当
时,
在
上单调递增;
(2)若
为奇函数,函数
,
,探究是否存在实数a,使
的最小值为
? 若存在,求出a的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49757d6d62b9c313b11aafd537475845.png)
(1)利用单调性定义证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48adb8a59b5c02fad5eada1b35171cf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/853ccd6cea4dd5f3491b10ca21828574.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c7b531ca02bb032e63ab7df9ee9e068.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
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解题方法
4 . 已知函数
,
,其中
.
(1)判断并证明
的单调性;
(2)①设
,
,求
的取值范围,并把
表示为
的函数
;
②若对任意的
,总存在
使得
成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44974d9d44f337b5901acb0389090234.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c3952b90cc044546c9315b47dfd460c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ac9eb4f13a6ec140f7050e8d7dde52c.png)
(1)判断并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)①设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1c9780f88088b5987da463a7b786aea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0a68eadbcb9953c6d7fc17ef2763ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0225bca34eaf19544939b29153aac1.png)
②若对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5fc4b242b118b3ba3a246d337cdc834.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b72b8620f1ccc3617f6a9e7ab366acb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23cbc24e5f8a7af7cdb12fafb890877a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
解题方法
5 . 已知函数
为奇函数.
(1)求实数
的值;
(2)判断函数
的单调性,并用函数单调性的定义证明;
(3)若存在
,使
在区间
上的值域为
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5291b459bed79d393360d029a4d0226.png)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(3)若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26701ee84c5cf514272fe188a34ae8ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8401162537a63e6c48c066e9f5fcdf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7056e33ad24df23ed625ce14d7c165d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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名校
6 . 若函数
与
满足:对任意
,都有
,则称函数
是函数
在集合上的“约束函数”.已知函数
是函数
在集合
上的“约束函数”.
(1)若
,
,判断函数
的奇偶性,并说明理由;
(2)若
,
,
,求实数a的取值范围;
(3)若
为严格减函数,
,
,且函数
的图象是连续曲线,求证:
是
上的严格增函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbc1bc250c8a6523a1be394ff48d4a51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ec6c7a1da7ecaef51a3d08fbcdf2821.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/facd6be947e37552dfa0565d1f21e380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f032c48bf8a18658be552c8fcd7f9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb635f785541935edc8bef1c30ba5483.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/526311a804ceb6d4b636447b02c750af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4d0a51632e4821be8823927b56ff038.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01bc9c32ab68ddb51b1a4196f50081f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f032c48bf8a18658be552c8fcd7f9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
您最近一年使用:0次
7 . 已知函数
的定义域为
,且
,
,都有
成立.
(1)求
,
的值,并判断
的奇偶性.
(2)已知函数
,当
时,
.
(i)判断
在
上的单调性;
(ii)若
均有
,求满足条件的最小的正整数
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a28068770a85b88b42321cd71ecd3c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df6593a700bf3e89107556454666b787.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8585e70c97056ae4c9f13b8594479758.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/649df22b06d6aa4123fcbbba8e759532.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74156327e5659301f391814605688899.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b530377e3fe56b7988935dd73d9dccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6e28dbfcdd6fb66b9ff759be044287.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fde64f4d3c38e43fbdee24eadc4b0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f36f7ab55b63c08280a41fb64366b819.png)
(i)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2052130268c701b5bc83f51dfe09958.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3832aedb9a10247ced52cb2fd89b7af9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
解题方法
8 . 已知函数
.
(1)判断函数
的奇偶性;
(2)判断函数
的单调性并证明;
(3)若
对任意的
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/362bfce584209628bc4ad3f23e3d7b11.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892e23f6d6a845ac8ee4c9f84fe66ed8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2fb40a36a293471742ce75f6b9635b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
2024-01-05更新
|
708次组卷
|
3卷引用:高一上学期期末数学考试模拟卷-【题型分类归纳】(人教A版2019必修第一册)
(已下线)高一上学期期末数学考试模拟卷-【题型分类归纳】(人教A版2019必修第一册)湖北省咸宁市崇阳县第二高级中学2023-2024学年高一上学期数学模拟考试试题(一)云南省祥华教育集团2023-2024学年高一下学期5月联考数学试题
9 . 若函数
在定义域
上满足
,且
时
,定义域为
的
为偶函数.
(1)求证:函数
在定义域上单调递增.
(2)若在区间
上,
;
在
上的图象关于点
对称.
(i)求函数
和函数
在区间
上的解析式.
(ii)若关于x的不等式
,
对任意定义域内的
恒成立,求实数
存在时,
的最大值关于a的函数关系.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dea20bf4103d4a86ce2dedc8cbf73498.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/018857ec6e498113b3b12a730d9313da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d991a665834f1957063731202084570.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
(1)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)若在区间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40c01b3dea6d0449097da0edc9130ef2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b577bf976fc3acd92b4af89be960359f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e110165a664ac7a77e70a6a46078602b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53a948d2f7732d7f03e986c63712089b.png)
(i)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d991a665834f1957063731202084570.png)
(ii)若关于x的不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2846c1cedbe564d20873d2b4d6f426aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6232dc74b15e4acb0ac3482a1cbe6a52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/157416e0bb98baff8059b9ef0e123ab5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
您最近一年使用:0次
2023-12-14更新
|
940次组卷
|
6卷引用:辽宁省大连市2022-2023学年高一上学期期末数学模拟试题
辽宁省大连市2022-2023学年高一上学期期末数学模拟试题江西省上饶市广丰区丰溪中学2023-2024学年高一上学期期末模拟数学试题(已下线)高一上学期期末考试解答题压轴题50题专练-举一反三系列福建省福州市九师教学联盟2023-2024学年高一上学期1月联考数学试题(已下线)高一数学开学摸底考 01-人教A版2019必修第一册全册开学摸底考试卷山东省德州市万隆中英文高级中学2023-2024学年高二下学期6月月考数学试题
名校
10 . 已知定义域为
的函数
为奇函数.
(1)求函数解析式
(2)证明函数单调性
(3)若关于
的不等式
对任意的
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2c36b75e4dc267e04921a9d049edc9a.png)
(1)求函数解析式
(2)证明函数单调性
(3)若关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d1df4bbc7c5d2d23e20efac67771482.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/636289ad84b4a3a51095dd32ca201f94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
您最近一年使用:0次
2023-12-13更新
|
609次组卷
|
5卷引用:内蒙古赤峰市林西县第一中学2023-2024学年高一上学期期末测试数学试题(A)
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