\begin{frame}
  \frametitle{Functions}
  
  \begin{block}{}
    A \emph{function} $f$ from $D$ to $E$ is a rule that
    assigns to each element $x$ in a set $D$
    exactly one element, called $f(x)$, in a set $E$.
  \end{block}
  Visualizing functions as \emph{arrow diagrams}:
  \begin{minipage}{.55\textwidth}
  \begin{center}
    \begin{tikzpicture}[default]
      \draw [fill=cblue!20,draw=cdblue] (0cm,0cm) to[out=10,in=90] (1cm,-1cm) to[out=-90,in=-10] (0cm,-2cm) to[out=170,in=-90,looseness=1.5] (-.2cm,-1cm) to[out=90,in=190,looseness=1.5] (0,0);
      \node (D) at (0,-2.6cm) {$D$};
      \node (x) at (.2cm,-.4cm) {$a$}; 
      \node (a) at (.4cm,-1cm) {$b$}; 
      \node (z) at (.1cm,-1.7cm) {$z$}; 
      \begin{scope}[xshift=35mm]
      \draw [fill=cblue!20,draw=cdblue,rotate=-20] (0cm,-1cm) ellipse (1cm and 1.3cm);
      \node (E) at (-.5,-2.6cm) {$E$};
      \node (a') at (-.2cm,-.1cm) {$a$}; 
      \node (c') at (-.4cm,-.8cm) {$b$}; 
      \node (q') at (-.5cm,-1.7cm) {$d$}; 
      \node (p') at (-0cm,-1.2cm) {$c$}; 
      \end{scope}
      \begin{scope}[cdred,->,>=stealth,thick]
      \draw (x) to[bend left=20] (a');
      \draw (a) to[bend left=-10] (a');
      \draw (z) to[bend left=-20] (q');
      \draw [shorten >= 5mm, shorten <= 5mm] (D) to node [above,black] {$f$} (E);
      \end{scope}
    \end{tikzpicture}
  \end{center}
  \end{minipage}
  \begin{minipage}{.44\textwidth}
    \pause\pause\pause\pause
    \begin{exampleblock}{This example}
      \begin{itemize}
    \pause
        \item domain $D = \{\;a,b,z\;\}$
    \pause
        \item $E = \{\;a,b,c,d\;\}$
    \pause
        \item $f(a) = \pause a$
    \pause
        \item $f(b) = \pause a$
    \pause
        \item $f(z) = \pause d$
    \pause
        \item range $= \pause\{\;a,d\;\}$
      \end{itemize}
    \end{exampleblock}
  \end{minipage}

  \setcounter{beamerpauses}{1}  
  \pause
  Terminology:
  \begin{itemize}
    \item $f(x)$ is the value of $f$ at $x$
  \pause
    \item \emph{domain} of $f$ is the set $D$
  \pause
    \item \emph{range} of $f$ is the set of all possible values $f(x)$ for $x$ in $D$
  \end{itemize}
\end{frame}