How is called the class of functions whose inverse function

Sonia Ayers 2022-07-13 Answered
How is called the class of functions whose inverse function is a polynomial? Is there any study of such functions?
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Answers (1)

Zackery Harvey
Answered 2022-07-14 Author has 21 answers
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 + + 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 + + b 0
be an arbitrary polynomial. Its inverse, P 1 ( x ), satisfies
P ( P 1 ( x ) ) = b n P 1 ( x ) n + b n 1 P 1 ( x ) n 1 + + 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.
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