\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}