解题方法
1 . 已知定义域为
的函数
,对任意
恒有
.
(1)求证:当
时,
.
(2)若
,恒有
,求证:
必有反函数.
(3)设
是
的反函数,求证:
在其定义域内恒有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd29ef32d9bc2e32ef2b8639b57dc9a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25bea6d14c16f7c06e4e028f36131360.png)
(1)求证:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66692ec49a458f9e48c7315d03dfc37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ad4c3cb38a5ce9b06167ce7217453d6.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fde64f4d3c38e43fbdee24eadc4b0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a71baf6217604517fd98fa97d0f55b43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d6b59f4796a45963dea76b89c72bea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d6b59f4796a45963dea76b89c72bea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c75efd66493102acfe77edff8fd9db97.png)
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2 . 函数
的图象经过点
,
.
(1)求函数
;
(2)设
,
,问:是否存在实数p(
),使
在区间
上是减函数,且在区间
上是增函数?证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af8981b3b896bc0c9ae0cb699f94c1d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2599bfa462c966a4988436b8c8bb7b30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcd9218a657b17654c5d757a6f7dee9a.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68e48e58aca82f136d6f0cc5251fd2e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff43a92f35dd115e3f8a3f2dd973b7a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7be8524456ba4e9abb973da323c0c88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46be55c8f2760d6db125f46691a3de48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27df58608819f3260cededaf16eb9770.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d156bb96e4a831d3f7c6e338a7cbfd0.png)
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解题方法
3 . 设
为定义域为
的函数,对任意
,都满足:
,
,且当
时,
.
(1)请指出
在区间
上的奇偶性、单调区间、最大(小)值和零点,并运用相关定义证明你关于单调区间的结论;
(2)证明
是周期函数,并求其在区间
上的解析式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17a3556b7e9458ec54d4cf1de17a5649.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34ea832b11f5a84b9bf3020271480631.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f7dbb416ec1ff1984a724a4f48bf692.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b24e7a6926a39f7be015218876573b63.png)
(1)请指出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d188ec2580e273ce87e51653a2177ee.png)
(2)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd765b568d695310edf942421a1cf499.png)
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4 . 已知函数
,其中
,
为实数且
.
(1)当
时,根据定义证明
在
单调递增;
(2)求集合
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9021fa0cc7427f534fbf1ec1db79c4b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2c80c26a794a844127aae7dee87c93b.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f0d68648b10fce54dfc19c5ee60086d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c0070f5fd205c3c48a41bd865a7049b.png)
(2)求集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac3810dfe2b1eb718bae23bf0bab35e.png)
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名校
解题方法
5 . 已知函数
是定义在
上的奇函数,且
.
(1)判断函数
的单调性并用定义加以证明;
(2)求使
成立的实数m的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46941285c660c2d772024a89114b1ce1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d188ec2580e273ce87e51653a2177ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)求使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/710242e11ad43cf2a947d2f992489612.png)
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2023-11-24更新
|
328次组卷
|
3卷引用:安徽省阜阳市第一中学2023-2024学年高一上学期数学竞赛试题
名校
解题方法
6 . 已知函数
,定义域为
.
(1)写出函数
的奇偶性(无需证明),判断并用定义法证明函数
在
上的单调性;
(2)若
,都有
恒成立,求实数
的取值范围;
(3)解不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7d5adeebe138f4d90677afd1ad7ce61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/662046df9f87264672dafd60d92e057b.png)
(1)写出函数
![](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/0109d06b8be2e402b5ffbb0aeb501009.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb61c076c156542dd4105842eefbf382.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e442f10e63ad0dd3144ea73d3fa6dcf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)解不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc0fa1210c98789833af075795fca365.png)
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2023-11-09更新
|
282次组卷
|
2卷引用:浙江省温州市苍南中学2023-2024学年高一上学期数学家摇篮竞赛试题
7 . 称
是
的一个向往集合,当且仅当其满足如下两条性质:(1)任意
,
;(2)任意
和
,有
.任取
,称包含
的最小向往集合称为
的生成向往集合,记为
.
(1)求满足
的正整数
的值;
(2)对两个向往集合
,定义集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccbe16b433635b8bc25f303863807b70.png)
(i)证明:
仍然是向往集合,并求正整数
,满足
;
(ii)证明:如果
,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/160af7e0b1d01eec9b33474b4d067a76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2077e5032491293f8181c4fc3bcf360a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3ad11a8563df9a39fbe386f746f755c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8104c761c3fac71e51c9a17a154829ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f27e8b43153beb780aa92d61df4b0da4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60cfb0de87efce8d98d89106fd36f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8060d3a485605dd9fedb3c5ae089c24e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c8f38fd2a2457ab28745c41c0f6b0aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086eb439f6a1578fdba904825340772d.png)
(1)求满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c248f486fa233098501ba2a64422118.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)对两个向往集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0248166f5a50eb4fe7f8a02a2d8e397e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccbe16b433635b8bc25f303863807b70.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57a13c9838a7aa389c93dcbaf5ad0449.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deb92321829e1fa81061502157411cec.png)
(ii)证明:如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/528af17b6a22c9c808c4231ef395a0c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0161489025ecbc391b1c9affce57b930.png)
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8 . 已知
,
(1)若函数
与
在
时有相同的值域,求
的取值范围;
(2)若方程
在
上有两个不同的根
,求
的取值范围,并证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e14c2f347403fb0f36d1549d10ab9f4.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b1d358875baefff9736f2f31c2e28e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(2)若方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae8b0843577a84645d1887c7136e9305.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afa482d7bcaa385bfc3548b42a4bfb60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad80c4ba8c593c5edfb167ae4a5f50f5.png)
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名校
解题方法
9 . 若
是定义在
上的偶函数,当
时,
.
(1)求
的解析式;
(2)讨论
在
上的单调性,并用定义证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d028846b8614318fbf90387d13c75b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e541ea2f855f981c96207070683d388.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4544c8626f01deff908469a90504b2c7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5265d99095b635f62c7915298ec0e963.png)
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解题方法
10 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a7f673cf793364aad2543ee8ae06228.png)
(1)请在网格纸中画出
的简图,并写出函数的单调区间(无需证明);
(2)定义函数
在定义域内的
,若满足
,则称
为函数
的一阶不动点,简称不动点;若满足
,则称
为函数
的二阶不动点,简称稳定点.
①求函数
的不动点;
②求函数
的稳定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a7f673cf793364aad2543ee8ae06228.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/3dd251f6-1acf-44cf-b925-66705e04e25c.png?resizew=210)
(1)请在网格纸中画出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)定义函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1de4841073ba41dc0e7b976759c3cd4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a52dc0a7f95a39091a2f11d80cc8579f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a576aa37d6f504669b40b7b38cb92694.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
①求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
②求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
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