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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}