解:(1)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055488327.gif)
是“S-函数”,则存在常数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055519282.gif)
,使得 (a+x)(a-x)=b.
即x2=a2-b时,对xÎR恒成立.而x2=a2-b最多有两个解,矛盾,
因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055488327.gif)
不是“S-函数”.………………………………………………3分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316505555072.gif)
若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055831450.gif)
是“S-函数”,则存在常数a,b使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055847461.gif)
,
即存在常数对(a, 32a)满足.
因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055831450.gif)
是“S-函数”………………………………………………………6分
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055441506.gif)
是一个“S-函数”,设有序实数对(a, b)满足:
则tan(a-x)tan(a+x)=b恒成立.
当a=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055925532.gif)
时,tan(a-x)tan(a+x)= -cot2(x),不是常数.……………………7分
因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055956607.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056003597.gif)
,
则有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231650560181594.gif)
.
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056034863.gif)
恒成立. ……………………………9分
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231650560491156.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056065806.gif)
,
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056065597.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056096484.gif)
时,tan(a-x)tan(a+x)=cot2(a)=1.
因此满足
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055441506.gif)
是一个“S-函数”的常数(a, b)=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055457636.gif)
.…12分
(3) 函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055020275.gif)
是“S-函数”,且存在满足条件的有序实数对
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056159281.gif)
和
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056159265.gif)
,
于是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056174818.gif)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231650561901311.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056205753.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056221730.gif)
.……………………14分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231650562372119.gif)
.………16分
因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056268858.gif)
, …………………………………………17分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231650562832015.gif)
综上可知当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165056299549.gif)
时函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165054833270.gif)
的值域为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165055472347.gif)
.……………18分