I'm interested in homeomorphisms between the real numbers, $\mathbb{R}$, and the positive real numbers, $(0,\infty)$--where both spaces have the topology induced by the metric $d(x,y)=|x-y|$.
Here is a couple of easy examples, $f:(0,\infty)\rightarrow\mathbb{R}$, I could think of
$$f(x)=\ln(x),\quad\quad\quad f(x)=\frac{x^{\alpha}-\gamma }{x^\beta},$$
where $\alpha,\beta,\gamma>0$ and $\alpha>\beta$. What other homeomorphisms are there?
EDIT: The motivation for the above question is to search for initial value problems
$\dot{x}=f(x),\quad\quad x_0\in\mathbb{R}^n_{>0}$
whose solutions can be mapped `nicely' to those of a linear system of ODEs on $\mathbb{R}^n$
$\dot{z}=Az,\quad\quad z_0\in\mathbb{R}^n.$
That way the analysis of the former, can be done by simply studying the later (which, generally will be much easier).
[Posted before motivation was added. Still might be useful.]
To find all homeomorphism between $X$ and $Y$, find one, $f:X\to Y$, and then find all self-homeomorphisms $Y\to Y$. Then every homeomorphism $X\to Y$ can be written as $h\circ f$ for some homeomorphism $h:Y\to Y$.
A homeomorphism $\mathbb R\to \mathbb R$ is a strictly monotonic continuous function that is unbounded above and below.
For diffeomorphisms, you can do the same thing.