Invertible Functions

Definition: Invertible Functions

A function \(f : X \rightarrow Y\) is defined to be invertible, if there exists a function \(g : Y \rightarrow X\) such that
\(gof = I_X\) and \(fog = I_Y\).
The function \(g\) is called the inverse of \(f\) and is denoted by \(f^{-1}\).

Thus, if \(f\) is invertible, then \(f\) must be one-one and onto and conversely, if \(f\) is one-one and onto, then \(f\) must be invertible.
This fact significantly helps for proving a function \(f\) to be invertible by showing that \(f\) is one-one and onto, especially when the actual inverse of \(f\) is not to be determined.

Example
Let \(f : \mathbb{N} \rightarrow Y\) be a function defined as \(f(x) = 4x + 3\), where
\(Y = \{ y \in \mathbb{N} : y = 4x + 3 \text{ for some } x \in \mathbb{N} \}\).
Show that \(f\) is invertible. Find the inverse.