The inversion of the exponential trigonometrical function

In this activity we learn whether a function can be inversed or not. As an examble we inverse the exponential function `f` Let`f` be the enponential trigonometrical function given by `f(x)=e^{\cos x}` The Domain set of the function `f` is the set `R`, the set of Real numbers The Range set of the function`f` is the subset of positive Real numbers The `y`-axis interception of the function `f` is the point `(0,e)` There are not any `x`-axis interceptions points of the function `f` Definision If a function `f` given by the formula `f(x)=y` is mapping `x` to `y` then the inverse function maps `y` back to `x` Activity Try to inverse the exponential trigonometrical function `f` if it is possible Important Rule A function `f` can be inversed, if the function can be defined in the range set of `f` or if the function `f` can be defined in a subset as a restriction of the range set of`f` Answer Let `g` be the inverse of the exponential trigonometrical function `f,` `f(x)=e^{\cos x}` given by the algebraic type `g(x)=\arccos(\ln x)` The Domain set of the function`g``is\ the\ subset` `[1/e,e]`, a subset of the set of possitive Real numbers The Range set of the function`g\ is\ the\ subset``[0,\pi]`, a subset of the set of possitive Real numbers Try to prove the algebraic type of `g` See also "Inverse trigonometric functions" at the link below https://en.wikipedia.org/wiki/Inverse_trigonometric_functions

By Stella Seremetaki Teacher Reference Number 1788154