\begin{frame} \frametitle{Power Functions: Special Cases} \begin{exampleblock}{} We consider $x^{\frac{1}{n}}$ where $n$ is a positive integer: \begin{itemize} \item $f(x) = x^{\frac{1}{n}} = \sqrt[n]{x}$ is a \emph{root function} (square root for $n=2$) \end{itemize} \end{exampleblock} \medskip \begin{minipage}{.49\textwidth} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=0ex] \diagram{-1}{3}{-2}{2}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{-1,1} \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=0.00001:3,samples=100] (\x,{pow(\x,1/6)}); \draw[cred] plot[smooth,domain=0.00001:3,samples=100] (\x,{pow(\x,1/4)}); \draw[cblue] plot[smooth,domain=0.00001:3,samples=100] (\x,{pow(\x,1/2)}); \draw[cblue] (-1.1cm,-2.8cm) -- node [at end,right] {$x^{\frac{1}{2}}$} +(.8cm,0); \draw[cred] (0.4cm,-2.8cm) -- node [at end,right] {$x^{\frac{1}{4}}$} +(.8cm,0); \draw[cgreen] (1.9cm,-2.8cm) -- node [at end,right] {$x^{\frac{1}{6}}$} +(.8cm,0); \node (a) [include=black,minimum size=1mm] at (1,1) {}; \node[r=(a),yshift=-2mm] {$(1,1)$}; \end{scope} \end{tikzpicture} } \end{minipage} \begin{minipage}{.49\textwidth} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-2}{2}{-2}{2}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{-1,1} \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=0.00001:2,samples=100] (\x,{pow(\x,1/7)}); \draw[cgreen] plot[smooth,domain=-2:-0.00001,samples=100] (\x,{-pow(-\x,1/7)}); \draw[cred] plot[smooth,domain=0.00001:2,samples=100] (\x,{pow(\x,1/5)}); \draw[cred] plot[smooth,domain=-2:-0.00001,samples=100] (\x,{-pow(-\x,1/5)}); \draw[cblue] plot[smooth,domain=0.00001:2,samples=100] (\x,{pow(\x,1/3)}); \draw[cblue] plot[smooth,domain=-2:-0.00001,samples=100] (\x,{-pow(-\x,1/3)}); \draw[cblue] (-2.1cm,-2.8cm) -- node [at end,right] {$x^{\frac{1}{3}}$} +(.8cm,0); \draw[cred] (-.6cm,-2.8cm) -- node [at end,right] {$x^{\frac{1}{5}}$} +(.8cm,0); \draw[cgreen] (.9cm,-2.8cm) -- node [at end,right] {$x^{\frac{1}{7}}$} +(.8cm,0); \node (a) [include=black,minimum size=1mm] at (1,1) {}; \node[r=(a),yshift=-2mm] {$(1,1)$}; \node (b) [include=black,minimum size=1mm] at (-1,-1) {}; \node[l=(b),yshift=2mm] {$(-1,-1)$}; \end{scope} \end{tikzpicture} } \end{minipage} \pause\medskip \begin{itemize} \item For even $n$ the domain is $[0,\infty)$, the graph is similar to $\sqrt{x}$. \item For odd $n$ the domain is $\mathbb{R}$, the graph is similar to $\sqrt[3]{x}$. \end{itemize} \vspace{10cm} \end{frame}