设n为正整数,规定:fn(x)=
(1)解不等式:f(x)≤x; (2)设集合A={0,1,2},对任意x∈A,证明:f3(x)=x; (3)求f2008(
|
(1)①当0≤x≤1时,由2(1-x)≤x得,x≥
.2 3
∴
≤x≤1.2 3
②当1<x≤2时,因x-1≤x恒成立.
∴1<x≤2.
由①,②得,f(x)≤x的解集为{x|
≤x≤2}.2 3
(2)∵f(0)=2,f(1)=0,f(2)=1,
∴当x=0时,f3(0)=f(f(f(0)))=f(-f(2))=f(1)=0;
当x=1时,f3(1)=f(f(f(1)))=f(f(0))=f(2)=1;
当x=2时,f3(2)=f(f(f(2)))=f(f(1))=f(0)=2.
即对任意x∈A,恒有f3(x)=x.
(3)f1(
)=2(1-8 9
)=8 9
,2 9
f2(
)=f(f(8 9
))=f(8 9
)=2 9
,14 9
f3(
)=f(f2(8 9
))=f(8 9
)=14 9
-1=14 9
,5 9
f4(
)=f(f3(8 9
))=f(8 9
)=2(1-5 9
)=5 9
,8 9
一般地,f4k+r(
)=fr(8 9
)(k,r∈N).8 9
∴f2008(
)=f0(8 9
)=8 9 8 9