\begin{frame} \frametitle{Inverse Functions} The inverse of a one-to-one function can be defined as follows. \pause \begin{block}{} Let $f$ be a one-to-one function with domain $A$ and range $B$. \medskip Then its \emph{inverse function} $f^{-1}$ is defined by: \begin{talign} f^{-1}(y) = x \;\;\iff\;\; f(x) = y \end{talign} and has domain $B$ and range $A$. \end{block} \pause \begin{exampleblock}{} The inverse function of $f(x) = x^3$ is $f^{-1}(y) = y^{\frac{1}{3}}$: \begin{talign} f^{-1}(f(x)) = f^{-1}(x^3) = (x^3)^{\frac{1}{3}} = x \end{talign} \end{exampleblock} \pause\medskip \begin{block}{} We have the following \emph{cancellation equations}: \begin{talign} f^{-1}(f(x)) = x && \text{for all $x \in A$}\\ f(f^{-1}(y)) = y && \text{for all $y \in B$} \end{talign} \end{block} \end{frame}