Sonia Ayers

2022-07-13

How is called the class of functions whose inverse function is a polynomial? Is there any study of such functions?

Zackery Harvey

Beginner2022-07-14Added 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

${a}_{n}(x)f(x{)}^{n}+{a}_{n-1}(x)f(x{)}^{n-1}+\cdots +{a}_{0}(x)=0$

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

$P({P}^{-1}(x))={b}_{n}{P}^{-1}(x{)}^{n}+{b}_{n-1}{P}^{-1}(x{)}^{n-1}+\cdots +{b}_{0}=x.$

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 $x$.

${a}_{n}(x)f(x{)}^{n}+{a}_{n-1}(x)f(x{)}^{n-1}+\cdots +{a}_{0}(x)=0$

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

$P({P}^{-1}(x))={b}_{n}{P}^{-1}(x{)}^{n}+{b}_{n-1}{P}^{-1}(x{)}^{n-1}+\cdots +{b}_{0}=x.$

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 $x$.

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