Definitions
The word inverse is related to the word invert meaning to reverse, turn upside down, to do the opposite.
Instead of considering the inverses for individual inputs and outputs, one can think of the function as sending the whole set of inputs, the domain, to a set of outputs, the range. Let ƒ be a function whose domain is the set X, and whose range is the set Y. Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:
If ƒ is invertible, the function g is unique; in other words, there is exactly one function g satisfying this property (no more, no less). That function g is then called the inverse of ƒ, denoted by ƒ−1.
Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, in which case the inverse relation is the inverse function.
Not all functions have an inverse. For this rule to be applicable, each element y ∈ Y must correspond to no more than one x ∈ X; a function ƒ with this property is called one-to-one, or information-preserving, or an injection.
Read more about this topic: Inverse Function
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