I have a infinitely differentiable, bijective function $f:[0,1]\to[0,1]$, and I would like to approximate this function by a series of other functions $T_i$ (think Taylor) – with the conditions that
- all the approximants are also bijective and
- its inverse must be representable by elementary functions (addition, multiplication, exponentation, $\exp$, $\log$, $\sin$ etc.).
Taylor polynomials do not fit this bill since, as polynomials, root finding is not elementary.
Any hints here? Links to articles could suffice.