Science:Math Exam Resources/Courses/MATH220/December 2011/Question 01 (e)/Solution 1

A function ƒ has an inverse if and only if it is bijective. That is, it has an inverse if and only if it is

1. Injective (or one-to-one)
2. Surjective (or onto).

In such a case, for every y ∈ B, there is a unique x ∈ A such that ƒ(x) = y. Using this fact, we define ƒ^-1 by the rule

$\displaystyle f^{-1}(y)=x$

where x is the unique element of the set A such that ƒ(x) = y. From the injectivity and surjectivity, this is well defined. Moreover,

$f{\big (}f^{-1}(y){\big )}=f(x)=y$

and

$f^{-1}{\big (}f(x){\big )}=f^{-1}(y)=x$

and so this really is the inverse of ƒ.