Function Inversion: Criteria for Elementary Form

[NoobixCube]

What is the criteria to know whether a function may be inverted into an elementary functional form?

 

[HallsofIvy]

Define "elementary functional form".

Any function that is "one to one" from set A onto set B then can be inverted ito an inverse function from B to A.

Do you consider "ln(x)" and "e^x" "elementary functional forms"? Since a common definition of "ln(x)" is "the inverse of e^x (and if it is defined in other ways, e^x is defined as "the inverse of ln(x)"), I guess you would agree that any "elementary functional form" that has an inverse can be inverted into "elementary functional form".