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\begin{frame}
\frametitle{Algebraic Functions}

\vspace{-1ex}
\begin{block}{}
A function $f$ is called \emph{algebraic function} if it can be constructed
using algebraic operations (addition, subtraction, multiplication, division
and taking roots) starting with polynomials.
\end{block}
\pause
\begin{exampleblock}{}
\begin{malign}
f(x) &= \sqrt{x^2 + 1} &\quad&&
g(x) &= \frac{x^2 - 16x^2}{x + \sqrt{x}} + (x-2) \sqrt[3]{x+1} &
\end{malign}
\end{exampleblock}

\pause\medskip
\begin{minipage}{.33\textwidth}
\scalebox{.57}{
\begin{tikzpicture}[default,baseline=1cm]
\diagram{-3.5}{1.5}{-2}{3}{1}
\diagramannotatez
\diagramannotatex{-1,1}
\diagramannotatey{-1,1}
\begin{scope}[ultra thick]
\draw[cgreen] plot[smooth,domain=-3:1.5,samples=50] (\x,{\x*sqrt(\x+3)});
\end{scope}
\end{tikzpicture}
}\\\smallskip
\centerline{{\small $x\sqrt{x+3}$}}
\end{minipage}~%
\begin{minipage}{.33\textwidth}
\scalebox{.57}{
\begin{tikzpicture}[default,baseline=1cm,xscale=.26,yscale=.85]
{\def\diaborderx{1.5cm}
\diagram[5]{-10}{10}{-1}{5}{1}}
\diagramannotatez
\diagramannotatex{-5,5}
\diagramannotatey{1,2}
\begin{scope}[ultra thick]
\draw[cred] plot[smooth,domain=-10:-5,samples=50] (\x,{pow(pow(\x,2) - 25,1/4)}) to (-5,0);
\draw[cred] (5,0) -- plot[smooth,domain=5.00001:10,samples=50] (\x,{pow(pow(\x,2) - 25,1/4)});
\end{scope}
\end{tikzpicture}
}\\\smallskip
\centerline{{\small $\sqrt[4]{x^2 - 25}$}}
\end{minipage}~%
\begin{minipage}{.33\textwidth}
\scalebox{.57}{
\begin{tikzpicture}[default,baseline=1cm]
\diagram{-1}{4}{-1}{4}{1}
\diagramannotatez
\diagramannotatex{-1,1}
\diagramannotatey{-1,1}
\begin{scope}[ultra thick]
\draw[cblue] plot[smooth,domain=-.51:-0.01,samples=290] (\x,{pow(pow(-\x,2),1/3)*pow(\x-2,2)}) -- (0,0);
\draw[cblue] (0,0) -- plot[smooth,domain=0.01:3.35,samples=290] (\x,{pow(pow(\x,2),1/3)*pow(\x-2,2)});
\end{scope}
\end{tikzpicture}
}\\\smallskip
\centerline{{\small $x^{\frac{2}{3}}(x-2)^2$}}
\end{minipage}
\end{frame}