inverse functionSomethingRelatedToInjectiveFunction2003-08-23 03:00:292008-02-11 14:03:49Definitionbijectioninvertible functioninvertible% this is the default PlanetMath preamble. as your knowledge
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{\bf Definition}
Suppose $f:X\to Y$ is a function between sets $X$ and $Y$,
and suppose $f^{-1}:Y\to X$ is a mapping that satisfies
\begin{eqnarray*}
f^{-1}\circ f &=& \operatorname{id}_X, \\
f\circ f^{-1} &=& \operatorname{id}_Y,
\end{eqnarray*}
where $\operatorname{id}_A$ denotes the identity function on the set $A$.
Then $f^{-1}$ is called the \emph{inverse of} $f$,
or the \emph{inverse function of} $f$.
If $f$ has an inverse near a point $x\in X$, then $f$ is
\emph{invertible near $x$}. (That is, if there is a set $U$ containing $x$
such that the restriction of $f$ to $U$ is invertible, then $f$ is invertible
near $x$.) If $f$ is invertible near all $x\in X$, then
$f$ is \emph{invertible}.
\subsubsection*{Properties}
\begin{enumerate}
\item When an inverse function exists, it is unique.
\item The inverse function and the inverse image of a set coincide
in the following sense.
Suppose $f^{-1}(A)$ is the inverse image of a set $A\subset Y$
under a function $f:X\to Y$.
If $f$ is a bijection, then $f^{-1}(y)=f^{-1}(\{y\})$.
\item The inverse function of a function $f:X\to Y$ exists if and only
if $f$ is a bijection, that is, $f$ is an injection and a surjection.
\item A linear mapping between vector spaces is invertible if and only if
the determinant of the mapping is nonzero.
\item For differentiable functions between Euclidean spaces, the inverse function
theorem gives a necessary and sufficient condition for the inverse to exist.
This can be generalized to maps between Banach spaces which are differentiable
in the sense of Frechet.
\end{enumerate}
\subsubsection*{Remarks}
When $f$ is a linear mapping (for instance, a matrix), the term \emph{non-singular} is
also used as a synonym for invertible.