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\begin{frame}
  \frametitle{Functions as Machines}
  
  A function as a \emph{machine}:
  \begin{center}
    \begin{tikzpicture}[default,node distance=25mm]
      \node (x) {$x$ in $D$};
      \node (f) [right of=x,minimum height=8mm,minimum width=12mm] {$f$};
      \machine{f}
      \node (fx) [right of=f] {$f(x)$ in $E$};
      \begin{scope}[cdred,->,>=stealth,ultra thick]
      \draw (x) -- (f.west);
      \draw (f.east) -- (fx);
      \end{scope}
      \node [at=(x.south),anchor=north] {(input)};
      \node [at=(fx.south),anchor=north] {(output)};
    \end{tikzpicture}
  \end{center}
  \begin{itemize}
  \pause
    \item \emph{domain} = set of all possible inputs
  \pause
    \item \emph{range} = set of all possible outputs
  \end{itemize}
  \pause
  
  \begin{example}
  Square $f(x) = x^2$:\\
  \begin{itemize}
  \pause
    \item domain = $\mathbb{R}$
  \pause
    \item range = \pause $\{x \mid x \ge 0\}$ = \pause $[0,\infty)$
  \end{itemize}
  \medskip\pause
  Square root $f(x) = \sqrt{x}$\; (over real numbers):\\
  \begin{itemize}
  \pause
    \item domain = \pause$\{x \mid x \ge 0\}$ = $[0,\infty)$
%     \hfill\textcolor{gray}{($\sqrt{x}$ for $x< 0$ does not exist)}
  \pause
    \item range = \pause$\{x \mid x \ge 0\}$ = $[0,\infty)$
  \end{itemize}
  \end{example}
\end{frame}