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At least the defining equations of the inverse $f^{-1}$ of $f$ with $y=f^{-1}(z)$ $\ \ $ $f^{-1}(f(y))=z$ and $f(f^{-1}(y))=z$ cannot be solved for $y$ by only transforming the equations only by inverting the operations (functions) directly readable from this equations. That's because the inverse relation of a multiary univalued algebraic function is multivalued and therefore not a function and in particular not an allowed function. But it has to be proved if there are other functions $f^{-1}$ contained in the field generated by the functions that build the function expression of $f$.

At least the defining equations of the inverse $f^{-1}$ of $f$ with $y=f^{-1}(z)$ $\ \ $ $f^{-1}(f(y))=z$ and $f(f^{-1}(y))=z$ cannot be solved for $y$ by only transforming the equations only by inverting the operations (functions) directly readable from these equations. That's because the inverse relation of a multiary univalued algebraic function is multivalued and therefore not a function and in particular not an allowed function. But it has to be proved if there are other functions $f^{-1}$ contained in the field generated by the functions that build the function expression of $f$.

My guess is that a bijective (generalized) composition $f$ of algebraic functions together with some differentiable unary univalued transcendental functions for which only function expressions exist that contain multiary univalued algebraic functions doesn't has an inverse in the differential field generated by the functions that build the function expression of $f$.

My guess is that a bijective (generalized) composition $f$ of algebraic functions together with some differentiable unary univalued transcendental functions for which only function expressions exist that contain multiary univalued algebraic functions doesn't have an inverse in the differential field generated by the functions that build the function expression of $f$.

J. F. Ritt proved in [Ritt 1925] that if a bijective elementary function $f$ has an elementary inverse, then $f=\psi_n\circ\ ...\ \circ \psi_1$, where each $\psi_i$ is either an algebraic function of one variable or else $\exp$ or $ln$. That means, there exists a representation of $f$ as elementary function that doesn't contain multiary algebraic functions.

J. F. Ritt proved in [Ritt 1925] that if a bijective elementary function $f$ has an elementary inverse, then $f=\psi_n\circ\ ...\ \circ \psi_1$, where each $\psi_i$ is either an algebraic function of one variable or else $\exp$ or $\ln$. That means, there exists a representation of $f$ as elementary function that doesn't contain multiary algebraic functions.