Inverse of a Function

What is the inverse of a function?

If we consider a function $f$ defined from a set $A$ to another set $B$, then the inverse of the function$f$, denoted by $f^{-1}$, is a function defined from the set $B$ to the set $A$, such that for every element $a\in A$, $f^{-1}(f(a))=a$. Moreover, a function whose inverse exists is called an invertible function.

In general, a function can be or not be invertible. Hence, it is natural to seek, when a function is invertible.

When is a function invertible?

A function is invertible if and only if it is a one-to-one correspondence. We must present an argument for both sides to see why this statement is correct. Let’s see both sides one by one.

First, we must show that if a function $f: A\to B$ is invertible, then it is a one-to-one correspondence. Moreover, because $f$ is an invertible function, $f^{-1}$ exists. If the function $f$ is not one-to-one, that means $a_1$ and $a_2$ are distinct elements in $A$ and $b\in B$ such that $f(a_1)=f(a_2) = b$ ...