Given that $f(x)$ is a function that maps real positive number to real positive number, and $f(x)=(f(x^{-1}))^{-1}$, could you find all the possible $f(x)$?
I know that $f(x)=x^a$ satisfies these conditions for any value of $a$. Is this the only function that satisfies the conditions? If not how can find all the other functions that satisfy the conditions?
Answer
Call $g(x)=\ln\circ f\circ \exp$. You want exactly $g(-x)=-g(x)$. So, the functions that satisfy that condition are precisely the functions $f$ such that $f(x)=e^{g(\ln x)}$ for some odd function $g:\Bbb R\to\Bbb R$. For instance, $f(x)=e^{\sin \ln x}$.
No comments:
Post a Comment