\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}