1 . 定义
,A中元素称为x奇函数;
,B中元素称为y奇函数;
,C中元素称为双偶函数.例如∶
,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede41d28405605d0b035108fadee0cd1.png)
(1)在下面横线上填下列词的一个∶ “真包含” “真包含于”“相等”,A∩B C,并说明理由;
(2)若所有项系数均为正数的多项式函数g(x,y),满足g(x,y)∈C,且g(x,y)=g(y,x),则可以找到关于t的多项式函数h(t),使得当x>0、y>0时,g(x,y)≥h(xy), 且等号当x= y>0时取到,求这样的h(t);
(3)证明∶对任何函数f(x,y),x∈R,y∈R,均可得到如下分解∶
,其中
为x奇函数,
为y奇函数,
为双偶函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fea6d93cb5d605d21fe86b3a92796828.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35bd194f1a72c7faeca9f2dec1f9c647.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8fe01caeda263d0069d2c5fd31085b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1fdbbc4dcf07441a069f1fa481741d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14c988035a9522f8e8e7fda10038d07e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede41d28405605d0b035108fadee0cd1.png)
(1)在下面横线上填下列词的一个∶ “真包含” “真包含于”“相等”,A∩B C,并说明理由;
(2)若所有项系数均为正数的多项式函数g(x,y),满足g(x,y)∈C,且g(x,y)=g(y,x),则可以找到关于t的多项式函数h(t),使得当x>0、y>0时,g(x,y)≥h(xy), 且等号当x= y>0时取到,求这样的h(t);
(3)证明∶对任何函数f(x,y),x∈R,y∈R,均可得到如下分解∶
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94d60bc39fc16f8695207d73101581f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07fd0c1e0ec352f8a9ce8b0f92ac95e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78b72607614bd7bd527880556b91b41e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b15b2c29613f899e609962bebb393908.png)
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解题方法
2 . 已知函数
在
上有意义,且对任意
满足
.
(1)求
的值,判断
的奇偶性并证明你的结论;
(2)若
时,
,判断
在
的单调性,并说明理由.
(3)在(2)的条件下,请在以下两个问题中任选一个 作答:(如果两问都做,按①得分计入总分)
①若
,请问是否存在实数
,使得
恒成立,若存在,给出实数
的一个取值;若不存在,请说明理由.
②记
表示
两数中的较大值,若对于任意
,
,求实数
的取值范围?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c275d203295b989c129101d82e74ae01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0f97718f1472e11502eaa775b58bd05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/608fd0dfd30079f4337ef571571eb287.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6afeede1e920a57feb40fc0cd66b961a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7baa4f3372e6a0aa38056e0de3b0fb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6000b174147cec2de26041837aec1b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c275d203295b989c129101d82e74ae01.png)
(3)在(2)的条件下,请在以下两个问题中
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ad3411b2f63b59dafb6fccdacddd1fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f32f78e3f288a433f8ba3661e551af4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
②记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eabe40ebe23d91aa1447b9896b300f83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/287376282d8c04d267ec6add486853f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fd851ca08ce2b6224e9d5e9952cff60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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2021-12-12更新
|
917次组卷
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2卷引用:北京师范大学附属中学2021-2022学年高一上学期期中考试数学试题
名校
3 . 设
的定义域是
,在区间
上是严格减函数;且对任意
,
,若
,则
.
(1)求证:函数
是一个偶函数;
(2)求证:对于任意的
,
.
(3)若
,解不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d188ec2580e273ce87e51653a2177ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff565afbddafe8625ef376d7eb3fa649.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc2435121b2b68da22ba4662e5734c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/333cf846facfab1283527ebe48961a95.png)
(1)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
(2)求证:对于任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1591d4244dcf5539a4ae98f554e91e61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5967cc62862986840af4dd29df4bcc41.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24f8b235e47a99a065a102c259b81db4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3059047431efe07f36b3fb319f709a78.png)
您最近一年使用:0次
2021-11-26更新
|
1217次组卷
|
5卷引用:上海市复旦大学附属中学2021-2022学年高一上学期期中数学试题
上海市复旦大学附属中学2021-2022学年高一上学期期中数学试题(已下线)专题03 函数的概念与性质(练习)-2(已下线)上海高一上学期期中【压轴42题专练】(2)重庆市南开中学2022-2023学年高一上学期12月月考数学试题(已下线)专题3-6 抽象函数性质综合归类(2) - 【巅峰课堂】题型归纳与培优练
4 . 已知函数
,
的定义域均为
,且满足:①
,
;②
为偶函数,
;③
,
,
.
(1)求
的值,并证明:
为奇函数;
(2)
,且
,证明:
①
;
②
单调递增.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27c97d54952104950bfd7afc0176bbd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f107d87b09135ba6960ee7bb57a4df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6422b9c2e93a91fe9e39ce4d9dabb0fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be9c2610130975ce70228f3a7fed50ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f107d87b09135ba6960ee7bb57a4df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/105a7e969af3553f18591c141d2df4a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df6593a700bf3e89107556454666b787.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54dad48527a47eab4a5916ab0421cc71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3baf34ce7cc1cf7228df43831495a3e.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/036be5035a4ee1aa3e05167dccdcfbc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27c97d54952104950bfd7afc0176bbd0.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aae2c13ad91dae29cf4d9f794a8808dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d9e99f6619b586d60f4f97504d7f6e6.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27c97d54952104950bfd7afc0176bbd0.png)
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