"Class" of functions whose inverse, where defined, is the same "class"

Their domains aren't generally all of $\Bbb R$, but the set of Moebius transformations, functions of the form $$f(x) = \frac{ax + b}{cx + d} \quad \text{with}\quad ad - bc \neq 0,$$ are a wonderful group of functions whose inverses are also of that form (and subsume linear functions, by letting $c = 0$). They're also called "linear fractional transformations" (I thought it was conventional to call them linear fractional transformations when you were only considering real number inputs, but evidently I'm wrong about that).

Generally people allow complex inputs for Moebius transformations.