Inverse Functions

If ƒ is a function from A to B, then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip returns each element to itself. Not every function has an inverse; those that do are called invertible.

Let ƒ be a function whose domain is the set X, and whose range is the set Y. Then the inverse of ƒ is the function ƒ^–1 with domain Y and range X, defined by the following rule:

If y = f (x), then f^-1(y) = x

If ƒ is an invertible function with domain X and range Y, then

f^-1( f (x) ) = x, for every x ∈ X

f^-1( f (y) ) = y, for every y ∈ Y

These two statements are equivalent to the definition of the inverse.