What are some examples of functions taking values from $(-\infty, \infty)$ to an open interval $(a,b)$? I'm trying to write a single index model that possesses this property, and the most popular example I can think of is $\frac{\exp(X\beta)}{1+\exp(X\beta)}$ of which values are in $(0,1)$. I can modify it to take values between $(a,b)$ but just wonder about more creative ideas...
Edit: It would be great that the function is as smooth as possible, twice differentiable is ideal! and monotone. Bijection is a plus, but it must be hard.