monotonic mapping from $(-\infty,\infty)$ to $(0,\infty)$

Question

I'm looking for a monotonic mapping from $(-\infty,\infty)$ to $(0,\infty)$. The ones I can think of are $e^{x}$ and $\log(1+e^x)$, however they both seem to grow exponentially at the left hand side of the output range (say from 0 to 1).

Is there a monotonic (preferably differentiable) mapping that grows at a lower level initially?

Answer 1

Try $f(x) = x + \sqrt{x^2+1}$.

Answer 2

$f(x)=-1/x$ for $x\leq-1$ and $x+2$ for $x>-1$ seems to be OK.

Answer 3

The function $f(x) = (k\sin x)^2$ for $x\in[2k\pi, 2(k+1)\pi]$, $k$ an integer.