Definition:Inverse Mapping

Definition
If the inverse $f^{-1}$ of a mapping $f$ is itself a mapping, then it is called the inverse mapping of $f$.

Thus, from the definition of a mapping, for $f^{-1}$ to be the inverse mapping of $f$:


 * $\forall y \in T: \left({x_1, y}\right) \in f \land \left({x_2, y}\right) \in f \implies x_1 = x_2$

and


 * $\forall y \in T: \exists x \in S: \left({x, y}\right) \in f$

When $f^{-1}$ is a mapping, we say that $f$ has an inverse mapping.

Invertible Mapping
If $f$ has an inverse mapping, then $f$ is an invertible mapping.

Also see

 * Bijection iff Inverse is Bijection, where is shown that $f^{-1}$ is a mapping iff $f$ is a bijection, and that $f^{-1}$ is itself a bijection.


 * Definition:Inverse of Bijection


 * Left and Right Inverses of Mapping are Inverse Mapping, which some sources use as the definition of an inverse mapping.

where $I_S$ and $I_T$ are the identity mappings on $S$ and $T$.
 * Bijection Composite with Inverse, that $f^{-1}$ is the two-sided inverse of $f$, i.e.:
 * $f \circ f^{-1} = I_S$
 * $f^{-1} \circ f = I_T$

Some sources use this property as the definition of an inverse mapping.