\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}