Exercise 1.1.76

Question

\begin{enumerate}[label=(\alph*)]
        \item If the point (5, 3) is on the graph of an even function, what other point must also be on the graph?
        \item If the point (5, 3) is on the graph of an odd function, what other point must also be on the graph?
    \end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate} [label=(\alph*)]
    \item Since for any even function $f$
    \begin{equation}
        \forall x\in X:\quad f(x)=f(-x),
    \end{equation}
    (where $X$ is the domain of $f$) if the point $(5, 3)$ is on the graph of $f$, point $(-5,3)$ must belong to the graph of $f$.
    \item For every odd function $f$:
    \begin{equation}\label{odd}
        \forall x\in X:\quad -f(x)=f(-x),
    \end{equation}
    where $X$ is the domain of $f$. Since the point $(5, 3)$ is on the graph of $f,$ we can take $x=5,\ f(x)=3.$ Substituting these values to the equation \ref{odd}, we get:
    $$-3=f(-5).$$
    Therefore, point $(-5,-3)$ belongs to the graph of $f.$
\end{enumerate}

Exercise 1.1.76

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Exercise 1.1.76

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