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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Inverse Functions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Inverse Functions\label{Inverse functions}}

\subsubsection{\label{One-to-one function}One-to-One Function}

\begin{definition}
A function $f$ with domain $A$ is called a \textbf{one-to-one function} if
no two elements of $A$ have the same image; that is, 
\[
f(x_{1})\neq f(x_{2})\qquad \text{whenever }x_{1}\neq x_{2}
\]
\end{definition}

\subsubsection{\label{Horizontal line test}Horizontal Line Test}

A function is one-to-one if and only if no horizontal line intersects its
graph more than once.

\subsubsection{\QTR{BOXHEAD}{Inverse Function}\quad}

\begin{definition}
Let $f$ be a one-to-one function with domain $A$ and range $B$. Then its 
\textbf{inverse function} $f$ $^{-1}$ has domain $B$ and range $A$ and is
defined by 
\[
f^{-1}(y)=x\quad \Leftrightarrow \quad f(x)=y
\]
for any $y$ in $B$.
\end{definition}

\subsubsection{\QTR{BOXHEAD}{Finding an Inverse Function}}

Follow these three steps to find the inverse of a one-to-one function $%
y=f(x) $:

\begin{enumerate}
\item  Write $y=f(x)$.

\item  Solve this equation for $x$ in terms of $y$ (if possible).

\item  Interchange $x$ and $y$.
\end{enumerate}

\noindent The resulting equation is $y=f^{-1}(x)$.\bigskip

The inverse function satisfies the identities

\begin{eqnarray*}
f^{-1}\left( f(x)\right) &=&x\quad \text{for every }x\text{ in }A \\
f\left( f^{-1}(x)\right) &=&x\quad \text{for every }x\text{ in }B
\end{eqnarray*}
These two identities completely determine the function $f^{-1}$.

\subsubsection{Continuity of Inverse Function}

\begin{theorem}
If $f$ is a one-to-one continuous function defined on an interval, then its
inverse function $f^{-1}$ is also continuous.
\end{theorem}

\subsubsection{Differentiability of Inverse Function}

\begin{theorem}
If $f$ is a one-to-one differentiable function with inverse function $%
g=f^{-1}$ and $f^{\prime }(g(a))\neq 0$, then the inverse function is
differentiable at $a$ and 
\[
g^{\prime }(a)=\dfrac{1}{f^{\prime }(g(a))}
\]
\end{theorem}

\subsection{Related Scientific Notebook Topics}

{\small \qquad \hyperref{Inverse function in Scientific Notebook}{}{}{%
../../Help/DM5-2.tex#Inverse function}}

{\small \qquad \hyperref{Plotting inverses with implicit plots}{}{}{%
../../Help/DM6-4.tex#Inverse function (2)}}

{\small \qquad \hyperref{Plotting inverses with parametric plots}{}{}{%
../../Help/DM6-4.tex#Plotting cube root}}

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