Question

can someone teach me what is a inverse function. And how to work out with them...

Answer 1

If you have a bijective function (meaning it is surjective and one-to-one), it has a unique inverse mapping every image of a point to its preimage which, by assumption, consists of only one element.

Answer 2

the inverse of any fn exists only if the fn bijective in the respective domain .

Answer 3

Given a function f(x), its inverse f^(-1)(x) is defined by
f(f^(-1)(x)) = f^(-1)(f(x))congruentx.
Therefore, f(x) and f^(-1)(x) are reflections about the line y = x. In Mathematica, inverse functions are represented using InverseFunction[f].
As noted by Feynman, the notation f^(-1) x is unfortunate because it conflicts with the common interpretation of a superscripted quantity as indicating a power, i.e., f^(-1) x = (1/f) x = x/f. It is therefore important to keep in mind that the symbols sin^(-1) z, cos^(-1) z, etc., refer to the inverse sine, inverse cosine, etc., and not to 1/sinz = cscz, 1/cosz = secz, etc.
A function f admits an inverse function f^(-1) (i.e., (open curly double quote)f is invertible(close curly double quote)) iff it is bijective. However, inverse functions are commonly defined for elementary functions that are multivalued in the complex plane. In such cases, the inverse relation holds on some subset of the complex plane but, over the whole plane, either or both parts of the identity f^(-1)(f(z)) = f(f^(-1)(z)) = z may fail to hold. A few examples are illustrated above and in the following table. In the table 0<b<a and c element Z.