Injection

Definition: Injection

A function $f:X\to Y$ is called injective if for each $y\in Y$ there is only one $x\in X$ such that $y=f(x)$.

Definition: Inverse Function

The inverse function of an injection $f:X\to Y$ is the function $f^{-1}:f(X)\to X$ which to each $y\in Y$ assigns the $x\in X$ for which $y=f(x)$, i.e.

$$f^{-1}(f(x))=x$$