Exercise 1.1.79 (79-80)

Question

The graph of a function defined for $x > 0$ is given. Complete the graph for $x \leq 0$ to make (a) an even function and (b) an odd function.

Khagan Gulusoy · Accepted Answer

\begin{enumerate}
    \item [79.] \begin{enumerate}
        \item For any even function $f$ the following is true:
        $$\forall x\in X:\quad f(x)=f(-x), $$
        where $X$ is the domain of the function $f.$ In other words, its graph must be symmetric about the $y$-axis. Using this rule, we can complete the graph as follows:
        \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, unit vector ratio=1 1 1, xmin=-1.4, xmax=1.4, ymin=-0.2, ymax=1.3,ticks=none, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:1.4, samples=50, mark=none, color=red, ultra thick] {4.9*x-7.4*x^2+3.2*x^3}; 
         \addplot[domain=-1.4:0, samples=50, mark=none, color=red, ultra thick] {-3.2*x^3-7.4*x^2-4.9*x};
    \end{axis}
    \end{tikzpicture}
    \end{figure}

\item For any odd function the  following is true:
    $$\forall x\in X:\quad f(-x)=-f(x), $$
    where $X$ is the domain of the function $f.$ In other words, its graph must be symmetric about the origin. Using this rule, we complete the graph as follows:
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, unit vector ratio=1 1 1, xmin=-1.4, xmax=1.4, ymin=-1.3, ymax=1.3,ticks=none, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:1.4, samples=50, mark=none, color=red, ultra thick] {4.9*x-7.4*x^2+3.2*x^3}; 
         \addplot[domain=-1.4:0, samples=50, mark=none, color=red, ultra thick] {4.9*x+7.4*x^2+3.2*x^3};
    \end{axis}
    \end{tikzpicture}
    \end{figure}
    \end{enumerate}

\item[80.] \begin{enumerate}
        \item For any even function the following is true:
        $$\forall x\in X:\quad f(x)=f(-x), $$
        where $X$ is the domain of the function $f.$ In other words, its graph must be symmetric about the $y$-axis. Using this rule, we can complete the graph as follows:
        \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, unit vector ratio=1 1 1, xmin=-4.2, xmax=4.2, ymin=-3, ymax=7,ticks=none, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:4, samples=50, mark=none, color=red, ultra thick] {-(16*x^2)/3+4*x^3-(2*x^4)/3}; 
         \addplot[domain=-4:0, samples=50, mark=none, color=red, ultra thick] {-(16*x^2)/3-4*x^3-(2*x^4)/3};
    \end{axis}
    \end{tikzpicture}
    \end{figure}
    \item For any odd function the following is true:
    $$\forall x\in X:\quad f(-x)=-f(x), $$
    where $X$ is the domain of the function $f.$ In other words, its graph must be symmetric about the origin. Using this rule, we complete the graph as follows:
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, unit vector ratio=1 1 1, xmin=-4.2, xmax=4.2, ymin=-7, ymax=7,ticks=none, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:4, samples=50, mark=none, color=red, ultra thick] {-(16*x^2)/3+4*x^3-(2*x^4)/3}; 
         \addplot[domain=-4:0, samples=50, mark=none, color=red, ultra thick] {(16 *x^2)/3 + 4 *x^3 + (2 *x^4)/3};
         
    \end{axis}
    
    \end{tikzpicture}
    \end{figure}
    \end{enumerate} 
\end{enumerate}

Exercise 1.1.79 (79-80)

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Exercise 1.1.79 (79-80)

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