Question

设函数f(x)={2∧x,x≤0, log2 x,x＞0. 则f[f(x)]=

Answer 1

答：

当x<=0时，f(x)=2^x>=1>0
当0<x<=1时，f(x)=log2(x)<=0
当x>1时，f(x)=log2(x)>0

所以：
x<=0时，f[f(x)]=log2[f(x)]=log2(2^x)=x
0<x<=1时，f[f(x)]=2^[f(x)]=2^[log2(x)]=x
x>1时，f[f(x)]=log2[f(x)]=log2[log2(x)]

所以：
当x<=1时，f[f(x)]=x；
当x>1时，f[f(x)]=log2[log2(x)]。

Answer 2

f(x)={2∧x,x≤0, log₂ x,x＞0.
当x≤0时，f(x)=2^x∈(0,1]
∴f[f(x)]=log₂(2^x)=x
当0<x≤1时，f(x)=log₂x≤0
∴f[f(x)]=2^(log₂x)=x
当x>1时，f(x)=log₂x>0
∴f[f(x)]=log₂(log₂x)
∴
f[f(x)]= {x ,x≤1
{log₂(log₂x) ,x>1

Answer 3

f(x)=2∧x ,x≤0,
=log<2>x ,x＞0
if x≤0
f(x) = 2^x >0
f(f(x)) = log<2> 2^x = x

if 0<x< 1
f(x) = log<2>x <0
f(f(x)) = x

if 1≤x
f(x) = log<2>x > 0
f(f(x)) = log<2>( log<2>x )

ie
f(f(x)) = x ; x<1
=log<2>( log<2>x ) ; x>=1