题目
题型:不详难度:来源:
(1)求证:f(x)是周期为4的周期函数;
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155536-27765.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155536-30935.png)
解析
试题分析:(1)只需证明
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155536-58284.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155536-60502.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-19245.png)
即有
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-99998.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-63126.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-14966.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-63126.png)
从而由
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-15584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155536-58284.png)
(2)首先由函数f(x)是定义在R上的奇函数,得到f(0)=0.
根据x∈[-1,0)时,-x∈(0,1],f(x)=-f(-x)=
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155538-77860.png)
利用函数的周期性得到,x∈[-5,-4]时,函数f(x)的解析式.
试题解析:(1)证明:由函数f(x)的图象关于直线
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155536-60502.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-19245.png)
即有
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-99998.png)
又函数f(x)是定义在R上的奇函数,故有
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-14966.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-63126.png)
故
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155537-15584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155538-94130.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155538-59302.png)
(2)由函数f(x)是定义在R上的奇函数,可知f(0)=0.
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155538-59785.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155539-32116.png)
故
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155538-59785.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155539-55229.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155539-96963.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155539-37536.png)
从而,
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155539-96963.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155536-30935.png)
核心考点
试题【已知函数f(x)是定义在R上的奇函数,且f(x)的图象关于直线x=1对称.(1)求证:f(x)是周期为4的周期函数;(2)若(0<x≤1),求x∈[-5,】;主要考察你对函数的相关概念等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155448-48922.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155448-45152.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155449-66704.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155406-43727.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155406-22687.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155406-53473.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155406-50457.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155407-11093.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155407-41006.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155406-22687.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155346-36605.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155347-15059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155347-16378.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155347-75454.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155347-94974.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155348-80106.png)
A.1 | B.2 | C.3 | D.4 |
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155244-77524.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155244-48175.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155245-63102.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155245-24211.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155245-89932.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155124-24732.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155125-54565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155125-78148.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155125-50655.jpg)
(Ⅰ)设半圆的半径
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155125-20307.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155126-40615.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155126-31141.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155126-37899.png)
(Ⅱ)由于受运动场两侧看台限制,
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155126-31141.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155127-58053.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155126-31141.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204155127-95166.png)
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