用[x]表示不超过x的最大整数,如[1.8]=1.对于下面关于函数f(x)=(x-[x])2的四个命题:
①函数y=f(x)的定义域为R,值域为[0,1];
②函数y=f(x)的图象关于y轴对称;
③函数y=f(x)是周期函数,最小正周期为1;
④函数y=f(x)在(0,1)上是增函数.
其中正确命题的序号是 .(写出所有正确命题的序号)
【答案】
分析:根据题意,以此分析4个命题:通过函数y=x-[x]∈[0,1)的值域可知①是否正确,通过举反例f(-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/0.png)
)≠f(
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/1.png)
),可得②不正确,通过周期函数的定义可知③是否正确,化简函数在(0,1)上的解析式可知函数y=f(x)在(0,1)上的单调性,综合可得答案.
解答:解:由题意可知:y=x-[x]∈[0,1),∴函数f(x)=(x-[x])
2的最大值取不到1,故①不对;
∵f(-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/2.png)
)=[-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/3.png)
-(-1)]
2=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/4.png)
,f(
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/5.png)
)=(
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/6.png)
-0)
2=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/7.png)
,则f(-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/8.png)
)≠f(
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024181535421437590/SYS201310241815354214375013_DA/9.png)
)
∴函数y=f(x)的图象不关于y轴对称,故②不对;
又知函数每隔一个单位重复一次,f(x+1)=(x-1-[x+1])
2=f(x),所以函数是以1为周期的函数,故③正确;
在(0,1)上f(x)=(x-[x])
2=(x-0)
2=x
2,∴函数y=f(x)在(0,1)上是增函数,故④正确;
故答案为 ③④.
点评:本题考查的是分段函数知识和函数值域等性质的综合类问题.在解答的过程当中充分体现了分类讨论的思想、特值的思想以及问题转化的思想.值得同学们体会反思.