Consider $f(x) = 2x$ and $g(x)=4x$. Is there any homeomorphism $h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h\circ f=g\circ h$

Question

Consider $f(x) = 2x$ and $g(x)=4x$. Is there any homeomorphism $h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h\circ f=g\circ h$?
Well it means $h(2x)=4h(x)$ but I don't know what to do from here?

Answer 1

$h(x) = x^2$ for $x>0$ and $h(x) = -x^2$ for $x<0$ seems to work.