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%%% Primary Title: inverse function
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%%% Owner: matte
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\begin{document}




{\bf Definition}
Suppose $f:X\to Y$ is a \htmladdnormallink{function}{http://planetmath.org/encyclopedia/Range2.html} between sets $X$ and $Y$,
and suppose $f^{-1}:Y\to X$ is a \htmladdnormallink{mapping}{http://planetmath.org/encyclopedia/Map2.html} 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 \htmladdnormallink{identity function}{http://planetmath.org/encyclopedia/IdentityOperator.html} 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 \htmladdnormallink{near}{http://planetmath.org/encyclopedia/IndiscreteProximity.html} a \htmladdnormallink{point}{http://planetmath.org/encyclopedia/Point.html} $x\in X$, then $f$ is
\emph{invertible near $x$}. (That is, if there is a set $U$ containing $x$
such that the \htmladdnormallink{restriction}{http://planetmath.org/encyclopedia/Restriction.html} 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 \htmladdnormallink{inverse image}{http://planetmath.org/encyclopedia/InverseImage.html} 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 \htmladdnormallink{bijection}{http://planetmath.org/encyclopedia/BijectiveFunction.html}, 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 \htmladdnormallink{injection}{http://planetmath.org/encyclopedia/Embedding.html} and a \htmladdnormallink{surjection}{http://planetmath.org/encyclopedia/Surjection.html}.
\item A \htmladdnormallink{linear mapping}{http://planetmath.org/encyclopedia/LinearOperator.html} between \htmladdnormallink{vector spaces}{http://planetmath.org/encyclopedia/LinearSpace.html} is invertible if and only if
the \htmladdnormallink{determinant}{http://planetmath.org/encyclopedia/Determinant2.html} of the mapping is nonzero.
\item For \htmladdnormallink{differentiable functions}{http://planetmath.org/encyclopedia/ContinuouslyDifferentiable.html} between \htmladdnormallink{Euclidean spaces}{http://planetmath.org/encyclopedia/EuclideanPlane.html}, the \htmladdnormallink{inverse function
theorem}{http://planetmath.org/encyclopedia/InverseFunctionTheorem.html} gives a \htmladdnormallink{necessary and sufficient}{http://planetmath.org/encyclopedia/Sufficiency.html} condition for the inverse to exist.
This can be generalized to maps between \htmladdnormallink{Banach spaces}{http://planetmath.org/encyclopedia/BanachSpace.html} which are \htmladdnormallink{differentiable}{http://planetmath.org/encyclopedia/ContinuouslyDifferentiable.html}
in the sense of Frechet.
\end{enumerate}

\subsubsection*{Remarks}
When $f$ is a linear mapping (for instance, a \htmladdnormallink{matrix}{http://planetmath.org/encyclopedia/Order7.html}), the term \emph{non-singular} is
also used as a synonym for invertible.

\end{document}