\begin{frame} \frametitle{Inverse Functions} \begin{block}{} A function $g$ is the inverse of a function $f$ if \begin{talign} g(f(x)) = x \quad \text{for all $x$ in the domain of $f$} \end{talign} (and the domain of $g$ is the range of $f$). \end{block} \pause\medskip \begin{minipage}{.50\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) {}; \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) {}; \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] (p'); \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}{.49\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) {}; \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) {}; \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] (p'); \draw (z) to[bend left=-20] (q'); \draw [shorten >= 5mm, shorten <= 5mm] (D) to node [above,black] {$f^{-1}$} (E); \end{scope} \end{tikzpicture} \end{center} \end{minipage} \pause\bigskip A function $f$ has an inverse if and only if $f$ is one-to-one. \end{frame}