解题方法
1 . “函数
的图象关于点
对称”的充要条件是“对于函数
定义域内的任意
,都有
,若函数
的图象关于点
对称,且当
时,
(1)求
的值;
(2)设函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c6bca97bb7f9c3694541a0c3803107c.png)
①证明函数
的图象关于点
称;
②若对任意
,总存在
,使得
成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba7204f43679af6935e494c59d40c6ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/535c58b5a37a5016bfbde48c15b77a25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f77ba8b3cb02c27e2a207a27a5f77701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/094bdef5a5a692a6cb194c8a9fea7266.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e66748312d59956072c0cd1bc08b40b.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0689085c7c4484df61d0b18d60953f4b.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c6bca97bb7f9c3694541a0c3803107c.png)
①证明函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52fe874d253faa184f61b1a3d7de7fd5.png)
②若对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb4361b7baf57ec27b60ac4aa637e16a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc4b407d102c2ad9b3278877f4f73a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e63bbadc6250f7139836ede33205550.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2 . 已知函数
的图像过点
.
(1)求函数
的解析式并直接写出函数
的定义域和值域;
(2)求
的值并指出函数
的对称中心;
(3)用单调性定义证明:函数
在区间
上是减函数;
(4)求函数
在
上的最值;
(5)若把函数
定义在集合
上,使它的值域是
,直接写出集合
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd78f04e8351e7293ec1e2807ff0a760.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4559e8b5861ebbf7c0f5c6d9a819f97.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb5e6e1113068cf3320eca992ea39c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)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/e99e986a24c7a655a1d5ec7e7688fe82.png)
(4)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1a1c92c42188e3b2cb800d1186eab12.png)
(5)若把函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00078668e2c7ab136413bce337ef2517.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
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解题方法
3 . 已知二次函数
满足
.
(1)求
,
的值;
(2)求证:
的图像关于直线
对称;
(3)用单调性定义证明:函数
在区间
上是增函数;
(4)若函数
是奇函数,当
时,
.
(i)直接写出
的单调递减区间为_________;
(ii)求出
的解析式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bab93efd42a3054040ccff8adf697c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3749d9ddfb2908ac0ee444743fe72afd.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
(3)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/189b2da6c420bf8f8900002d14f65f72.png)
(4)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1d1a94ea3c278c2197572cc1b7725b1.png)
(i)直接写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(ii)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
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4 . 已知二次函数
,有两个零点为
和
.
(1)求
、
的值;
(2)证明:
;
(3)用单调性定义证明函数
在区间
上是增函数;
(4)求
在区间
上的最小值
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83ce0c1641cb63f8b725aed093f09d4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b3b657ebd1733b4f19dcbec44919924.png)
(3)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/e5d6243e93c41978871cb23d8e66148d.png)
(4)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c3e441923ed3c1a32720d6aeac2f599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01b3ae7e5228fd1acb0d46f6941143a7.png)
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