函数f(x)的定义域为R,且满足:1.对于任意的x,y属于R,f(x-y+1)=f(x)f(y)+f(1-x)f(1-
函数f(x)的定义域为R,且满足:1.对于任意的x,y属于R,f(x-y+1)=f(x)f(y)+f(1-x)f(1-y);2.f(x)在区间【0,1】上单调递增求(1)...
函数f(x)的定义域为R,且满足:1.对于任意的x,y属于R,f(x-y+1)=f(x)f(y)+f(1-x)f(1-y); 2.f(x)在区间【0,1】上单调递增求(1)f(0)(2)不等式2f(x+1)大于等于1的解集
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f(x-y+1)=f(x)f(y)+f(1-x)f(1-y)
f(x+(1-y))=f(x)f(1-(1-y))+f(1-x)f(1-y);
f(x+y)=f(x)f(1-y)+f(1-x)f(y) ,
设x=y=1/3,
f(1/3+1/3)=f(1/3)f(1-1/3)+f(1-1/3)f(1/3) ,
f(2/3)=f(1/3)f(2/3)+f(2/3)f(1/3) ,
f(2/3)=2f(1/3)f(2/3) ,
1/2=f(1/3)
[0,1]上单调递增,
f(1/3)=1/2,
f(2x-1)≥1/2=f(1/3),
设y=1, f(1+x)=f(1-x), [关于x=1对称,f(1/3)=f(5/3),]
即f(x)=f(2-x),是奇函数 ,
-f(x)=f(-x)=f(2+x),
f(0)=0;
f(4+x)=f[2+(2+x)]=-f(2+x)=f(x),
周期为4,
(0,2),f(x)>0;(2,4),f(x)<0,
(0,2),f(x)>1/2,1/3<x<5/3;x∈R,4k+1/3<x<4k+5/3.
f(x+1)>1/2 =>x∈R,4k+1/3<x+1<4k+5/3.=>4k-2/3<x<4k+2/3.
f(x+(1-y))=f(x)f(1-(1-y))+f(1-x)f(1-y);
f(x+y)=f(x)f(1-y)+f(1-x)f(y) ,
设x=y=1/3,
f(1/3+1/3)=f(1/3)f(1-1/3)+f(1-1/3)f(1/3) ,
f(2/3)=f(1/3)f(2/3)+f(2/3)f(1/3) ,
f(2/3)=2f(1/3)f(2/3) ,
1/2=f(1/3)
[0,1]上单调递增,
f(1/3)=1/2,
f(2x-1)≥1/2=f(1/3),
设y=1, f(1+x)=f(1-x), [关于x=1对称,f(1/3)=f(5/3),]
即f(x)=f(2-x),是奇函数 ,
-f(x)=f(-x)=f(2+x),
f(0)=0;
f(4+x)=f[2+(2+x)]=-f(2+x)=f(x),
周期为4,
(0,2),f(x)>0;(2,4),f(x)<0,
(0,2),f(x)>1/2,1/3<x<5/3;x∈R,4k+1/3<x<4k+5/3.
f(x+1)>1/2 =>x∈R,4k+1/3<x+1<4k+5/3.=>4k-2/3<x<4k+2/3.
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