The inverse of a function f is denoted by f-1 and it exists only when f is both one-one and onto function. Note that f-1 is NOT the reciprocal of f. The composition of the function f and the reciprocal function f-1 gives the domain value of x.
(f o f-1) (x) = (f-1 o f) (x) = x
For a function ‘f’ to be considered an inverse function, each element in the range y ∈ Y has been mapped from some element x ∈ X in the domain set, and such a relation is called a one-one relation or an injunction relation. Also the inverse f-1 of the given function has a domain y ∈ Y is related to a distinct element x ∈ X in the codomain set, and this kind of relationship with reference to the given function ‘f’ is an onto function or a surjection function. Thus the inverse function being an injunctive and a surjection function, is called a bijective function.
Let us consider a function f whose domain is the set X and the codomain is the set Y. The function f is invertible if there exists another function g whose domain is Y and its codomain is X. These two functions can be represented as f(x) = Y, and g(y) = X. For this situation, if the function f(x) is inverse, then its inverse function g(x) is unique.
If the composition of two functions f(x), and g(x), results in an identity function f(g(x))= x, then the two functions are said to be inverses of each other. If the application of a function to x as input gives n output of y, then the application of another function g to y should give back the value of x. Hence the inverse of a function reverses the function. The domain of the given function becomes the range of the inverse function, and the range of the given function becomes the domain of the inverse function.