(1)解:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180358768767.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180358783764.gif)
2分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180358799586.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231803588301602.gif)
3分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180358846389.gif)
是偶函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082318035903372.gif)
4分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359080757.gif)
5分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082318035903372.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180358846389.gif)
是非奇非偶函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082318035903372.gif)
6分
(2)(理)方法一(积化和差):
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359142607.gif)
为偶函数,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231803591581856.gif)
8分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359142607.gif)
为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082318035925172.gif)
偶函数,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359267698.gif)
是偶函数, 9分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359283519.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359454705.gif)
10分
方法二(定义法):
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359142607.gif)
为偶函数
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231803594851618.gif)
展开整理
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359501876.gif)
对一切
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180357941247.gif)
恒成立 8分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359548454.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359563674.gif)
10分
方法三(特殊值法):
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359142607.gif)
为偶函数
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231803596411559.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180359860631.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231804000001549.gif)
8分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400016616.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400063684.gif)
10分
(文)、方法一(定义法):
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400078791.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400094889.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180358846389.gif)
偶函数,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400141523.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400172664.gif)
, 8分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400187458.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400063684.gif)
10分
方法二(特殊值法):
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400234599.gif)
为偶函数
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231804002501625.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400281609.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231804002971624.gif)
8分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400312488.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400328656.gif)
10分
(3)第一层次,写出任何一种的一个(加法或乘法)均可以, 13分
1、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400515482.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400531501.gif)
是偶函数;
2、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400546488.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400531501.gif)
是奇函数;
3、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400655476.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400531501.gif)
是非奇非偶函数;
4、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400811479.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400531501.gif)
既奇又偶函数
第二层次,写出任何一种的一个(加法或乘法)均可以, 14分
1、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400967483.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401108540.gif)
是偶函数 (数字不分奇偶)
2、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400546488.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401233541.gif)
是奇函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400546488.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401264553.gif)
是偶函数
(数字只能同奇数)
3、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400655476.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401233541.gif)
是非奇非偶函数 (数字不分奇偶,但需相同)
4、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400811479.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401108540.gif)
是既奇又偶函数 (数字只能奇数)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400811479.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082318040198172.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401997544.gif)
是非奇非偶函数
第三层次,写出逆命题任何一种的一个(加法或乘法)均可以, 15分
1、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401108540.gif)
是偶函数 (数字不分奇偶,但相同), 则
2、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401233541.gif)
是奇函数 (数字只能正奇数) ,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402325479.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401997544.gif)
是偶函数 (数字只能正偶数), 则
3、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401108540.gif)
是偶函数 (数字只能正奇数),则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402387471.gif)
第四层次,写出充要条件中的任何一种均可以, 16分
1、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402075472.gif)
的充要条件是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180400531501.gif)
是偶函数
2、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401233541.gif)
是奇函数 (数字只能正奇数) 的充要条件是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402325479.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401997544.gif)
是偶函数 (数字只能正偶数)的充要条件是
3、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180401108540.gif)
是偶函数 (数字只能正奇数)的充要条件是 则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402387471.gif)
第五层次,写出任何一种均可以(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082318040294965.gif)
逆命题,充要条件等均可以,限于篇幅省略) 18分
1、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402964603.gif)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402995564.gif)
都是偶函数
2、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180403058601.gif)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180403073192.gif)
是正奇数,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402995564.gif)
是奇函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180403058601.gif)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180403073192.gif)
是正偶数,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402995564.gif)
是偶函数
3、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180403183590.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180403073192.gif)
奇数,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402995564.gif)
既奇又偶函数
4、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180403183590.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180403073192.gif)
偶数,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823180402995564.gif)
是非奇非偶函数