函数f(x^2-1)=logm(x^2/(2-x^2)),m>0,m不等于1

令a=x^2-1
x^2=a+1

f(a)=logm[(a+1)/(1-a)]
f(x)=logm[(x+1)/(1-x)]
定义域
(x+1)/(1-x)>0
(x+1)(1-x)>0
(x+1)(x-1)<0
-1<x<1
定义域关于原点对称
可以讨论奇偶性

f(x)+f(-x)
=logm[(x+1)/(1-x)]+logm[(-x+1)/(1+x)]
=logm{[(x+1)/(1-x)][(-x+1)/(1+x)]}
=logm(1)
=0
f(-x)=-f(x)
奇函数

logm[(x+1)/(1-x)]≥logm(3x+1)
若0<m<1
则(x+1)/(1-x)≤3x+1
(x+1)/(1-x)-(3x+1)≤0
[(x+1)-(3x+1)(1-x)]/(1-x)≤0
(3x^2-x)/(x-1)≥0
x(x-1)(3x-1)≥0
x≥1,0≤x≤1/3
加上定义域
0≤x≤1/3

若m>1
则(x+1)/(1-x)≥3x+1>0
(x+1)/(1-x)-(3x+1)≥0
[(x+1)-(3x+1)(1-x)]/(1-x)≥0
(3x^2-x)/(x-1)≤0
x(x-1)(3x-1)≤0
x≤0,1/3≤x≤1
3x+1>0
x>-1/3
加上定义域
-1/3<x≤0或1/3≤x<1