\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}