Exercise 1.1.77 (77-78)

Question

Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

Khagan Gulusoy · Accepted Answer

\begin{enumerate}
    \item [77.] \begin{itemize}
        \item The graph of the function $f$ is symmetric about the origin; therefore,
        \begin{equation}\label{oddprop}
            \forall x\in \mathbb{X}:\quad f(-x)=-f(x),
        \end{equation}
                where $\mathbb{X}$ is the domain of $f$. Therefore, in this example, $f$ is an odd function.
        \item The graph of the function $g$ is symmetric with respect to the $y$-axis; therefore,
        \begin{equation}\label{evenprop}
            \forall x\in \mathbb{X}:\quad g(-x)=g(x),
        \end{equation} where $\mathbb{X}$ is the domain of $f$. Therefore, $g$ is even.

\end{itemize}
    \item[78.] \begin{itemize}
        \item Since for the function $f$ none of the properties \ref{oddprop} and \ref{evenprop} are true, i.e., $f$ is neither symmetric about the origin nor with respect to the $y$-axis, we can conclude that $f$ is neither even nor odd.
        \item Function $g$ is even because its graph is symmetric with respect to the $y$-axis, i.e., the property \ref{evenprop} is true for the function $g.$

\end{itemize} 
\end{enumerate}

Exercise 1.1.77 (77-78)

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Exercise 1.1.77 (77-78)

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