How is called the class of functions whose inverse function is a polynomial? How is called the class of functions whose inverse function is a polynomial? Is there any study of such functions?
 A: The functions you're interested in are a particular subset of the algebraic functions.  To be specific, an algebraic function $f(x)$ is a function that satisfies
$$
a_n(x)f(x)^n+a_{n-1}(x)f(x)^{n-1}+\cdots+a_0(x)=0
$$
where the $a_i(x)$ are polynomials.  By contrast, let $$P(x)=b_n x^n + b_{n-1}x^{n-1} + \cdots + b_0$$ be an arbitrary polynomial.  Its inverse, $P^{-1}(x)$, satisfies
$$
P(P^{-1}(x))=b_nP^{-1}(x)^n+b_{n-1}P^{-1}(x)^{n-1}+\cdots+b_0=x.
$$
So inverses of polynomials are those algebraic functions for which the defining coefficient functions $a_i$ are constants, except for the last, which is a constant minus $x$.  
A: Unless we want a restriction that the inverse be "globally" a polynomial, i.e. the inverse of a monotone polynomial to ensure the requirement of being a function, the phrase "inverse polynomial" will probably convey the intended class of functions well enough.
Interpolation by inverse polynomials, esp. inverse quadratic interpolation, has been studied in connection with root-finding and line searches in optimization.
For degree 1 the inverse polynomial doesn't yield any new methods.  For degree 2 the inverse (quadratic) polynomial has found a popular niche in Brent's algorithm.
