【答案】
分析:(1)由于 函数f(x)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/0.png)
是定义在实数集R上的奇函数,则f(-x)=-f(x),构造方程,可求a与b值;
(2)由题意以及①当x∈[0,3)时,g(x)=f(x);②g(x+3)=g(x)lnm(m≠1).得到
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/1.png)
;
对参数lnm分类讨论,再依据函数g(x)在x∈[0,+∞)上的值域是闭区间,即可得到m的取值范围.
解答:解:(1)由函数f(x)定义域为R,∴b>0.
又f(x)为奇函数,则f(-x)=-f(x)对x∈R恒成立,得a=0.(2分)
因为y=f(x)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/2.png)
的定义域为R,所以方程yx
2-x+by=0在R上有解.
当y≠0时,由△≥0,得-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/3.png)
≤y≤
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/4.png)
,
而f(x)的值域为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/5.png)
,所以
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/6.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/7.png)
,解得b=4;
当y=0时,得x=0,可知b=4符合题意.所以b=4.(5分)
(2)①因为当x∈[0,3)时,g(x)=f(x)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/8.png)
,
所以当x∈[3,6)时,g(x)=g(x-3)lnm=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/9.png)
;(6分)
当x∈[6,9)时,g(x)=g(x-6)(lnm)
2=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/10.png)
,
故
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/11.png)
(9分)
②因为当x∈[0,3)时,g(x)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/12.png)
在x=2处取得最大值为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/13.png)
,在x=0处取得最小值为0,(10分)
所以当3n≤x<3n+3(n≥0,n∈Z)时,g(x)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/14.png)
分别在x=3n+2和x=3n处取得最值为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/15.png)
与0.(11分)
(ⅰ) 当|lnm|>1时,g(6n+2)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/16.png)
的值趋向无穷大,从而g(x)的值域不为闭区间;(12分)
(ⅱ) 当lnm=1时,由g(x+3)=g(x)得g(x)是以3为周期的函数,从而g(x)的值域为闭区间
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/17.png)
;(13分)
(ⅲ) 当lnm=-1时,由g(x+3)=-g(x)得g(x+6)=g(x),得g(x)是以6为周期的函数,
且当x∈[3,6)时g(x)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/18.png)
值域为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/19.png)
,从而g(x)的值域为闭区间
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/20.png)
;(14分)
(ⅳ) 当0<lnm<1时,由g(3n+2)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/21.png)
<
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/22.png)
,得g(x)的值域为闭区间
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/23.png)
;(15分)
(ⅴ) 当-1<lnm<0时,由
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/24.png)
≤g(3n+2)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/25.png)
<
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/26.png)
,从而g(x)的值域为闭区间
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/27.png)
.
综上知,当m∈
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125657960628370/SYS201310251256579606283018_DA/28.png)
∪(1,e],即0<lnm≤1或-1≤lnm<0时,g(x)的值域为闭区间.(16分)
点评:本题考查的知识点是函数奇偶性,函数的值域,解题的关键是熟练掌握函数奇偶性的性质,以及分类讨论求出参数的取值范围.