# Inverse Functions

Suppose a set of ordered pairs defines a relation. Then the inverse relation is obtained by reversing the coordinates of each ordered pair. This means if (a, b) is an element of a relation then (b, a) is the corresponding element in the inverse relation. In short inverse relation switches input and output. If both the relation and its inverse are functions then two functions serve as inverse functions of each other.

The function g which is the inverse of f can be denoted as f^{-1} and read as "f inverse".

The
Domain and Range of function switch the roles as Range and Domain of
the inverse function. This means all functions do not have inverses.
For an inverse function to be defined the original function needs to be
one-to-one, that is no image in the range can have more than one pre image in the domain.__ Definition:__ Suppose f(x) is an one-to-one function with domain D and range R. Then its inverse function f

^{-1}(x) has domain R and range D and defined by

f

^{-1}(y) = x ⇔ f(x) = y for any y ∈ R.