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,

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