Question

已知定义域为R的函数f(x)=(-2^x+b)/(2^x+1)是奇函数。（1）求b的值。（2）判断函数f(x)的单调性。并给予证

Answer 1

f(-x)=(-1/2^x+b)/(1/2^x+1)=-f(x)=(2^x-b)/(2^x+1)
(-1+b2^x)/(2^x+1)=(2^x-b)/(2^x+1)
-1+b2^x=2^x-b
b(2^x+1)=2^x+1 b=1
f(x)=(-2^x+b)/(2^x+1)=f(x)=(1-2^x)/(2^x+1)=(2-1-2^x)/(2^x+1)=2/(2^x+1)-1=1/(2^(x-1)+1/2)-1
函数f(x)为减函数
x<x+1
1/(2^(x-1)+1/2)-1-1/(2^x+1/2)+1=1/(2^(x-1)+1/2)-1/(2^x+1/2)=(2^x-2^(x-1))/(2^(x-1)+1/2)*(2^x+1/2)=2^(x-1)/(2^(x-1)+1/2)*(2^x+1/2)>0 f(x)<f(x+1)