Introduction
- If f is a function, then its inverse function g is defined as the reverse of the mapping created by f.
- If g is an inverse of f then:
- and:
Example
- To find the inverse of a function given as an equation, we can simply solve for y
Notation
To denote that a function is an inverse of another, two different notations are commonly used:- If the function is representedby the letter f then its inverse is can be denoted using a superscript of -1
LaTeX:
f^{-1}(x) - If the function is represented by the letter f then its inverse is can be denoted using the letter g
LaTeX:
g(x)
Properties
- If the original function is a bijection then its inverse will also be a bijection.
- If the original function is not a bijection then there is no guarantee that it has an inverse.
- If it is not injective then there may be two elements in the domain that map to the same element in the codomain.
- If it is not surjective then there may be an element in the codomain that is not mapped to from the domain.