一道复合函数题目!设f(x)=a^x^2 (a>0,a不等于1）,f[g(x)]=1-x,且g(x)>=0,求g(x).

Question

一道复合函数题目!
设f(x)=a^x^2 (a>0,a不等于1）,f[g(x)]=1-x,且g(x)>=0,求g(x).

Answer 1

设f(x)=a^(x^2) (a>0,a不等于1）,f[g(x)]=1-x,且g(x)>=0,求g(x).
设y=g(x)
1-x=f[g(x)]=f(y)=a^(y^2)=a^{[g(x)]^2},即
a^{[g(x)]^2}=1-x,两端取以a为底的对数,-->
[g(x)]^2=log(1-x),-->
g(x)=√[log(1-x)]
或g(x)=-√[log(1-x)]