How to construct $$f: \mathbb{R} \mapsto (a,b)$$?

There are three different functions that I can thought of: Firstly is the inverse tangent. Since tangent maps $$[0,\pi]$$ to $$\mathbb{R}$$, its inverse serves the purpose.

Second function is the similar hyperbolic tangent, defined as

$\mathrm{tanh}(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}.$

This function maps $$\mathbb{R}$$ into $$(-1,1)$$.

The third function is the logistic function:

$P(x) = \frac{1}{1+e^{-x}},$

which maps $$\mathbb{R}$$ into $$(0,1)$$. By the way, logistic function is named because it is the solution to the logistic equation,

$\frac{d}{dx}P(x) = P(x)(1-P(x)).$