Even and Odd Functions

\subsection{Even and Odd Functions}

For a function $f$ in the form $y=f(x)$, we describe its type of symmetry by
calling the function \textbf{even}\index{even functions} or
\textbf{odd}\index{odd functions}.

An \textbf{even function} means $f(-x)=f(x)$.
An example of an even function is the function $f(x)=x^2$.
  \begin{figure}[H]
    \begin{center}
      \begin{tikzpicture}
        \begin{axis}[
            ylabel={$f(x)=x^2$},
            axis x line=bottom,
            axis y line=center,
            tick align=outside,
            yticklabels={,,}
            xticklabels={,,}
            xtickmax=10,
          ]
          \addplot[smooth,red]{x^2};
        \end{axis}
      \end{tikzpicture}
    \end{center}
    \caption{$f(x)=x^2$ is an \emph{even function}.}
  \end{figure}
  An \textbf{odd function} means $f(-x)=-f(x)$. An example of this is the
  function $f(x)=x^3$.
  \begin{figure}[H]
    \begin{center}
      \begin{tikzpicture}
        \begin{axis}[
            ylabel={$f(x)=x^3$},
            axis x line=bottom,
            axis y line=center,
            tick align=outside,
            yticklabels={,,}
            xticklabels={,,}
            xtickmax=10,
          ]
          \addplot[smooth,red]{x^3};
        \end{axis}
      \end{tikzpicture}
    \end{center}
    \caption{$f(x)=x^3$ is an \emph{odd function}.}
  \end{figure}
\subsection{Surjective, Injective, and Bijective Functions}

  \index{one-to-one}
  \index{injective}
  If each $f(x)$ value produced by a function $f$ can only be obtained by one
  unique $x$ value, then we say $f$ is \textbf{injective}, or
  \textbf{one-to-one}.

  $ f: D \to R $ is injective or one-to-one iff
  \[
    \forall{(x_1 \wedge x_2 \in D)}
    \big[f(x_1)=f(x_2)
    \to x_1=x_2\big].
  \]
  \begin{remark}
    This also means that for injective functions,
    $ x_1 \neq x_2 \to f(x_1) \neq f(x_2)$.
  \end{remark}

\begin{figure}[H]
    \begin{center}
        \subfigure[The function $f(x)=x^2$ is not \emph{one-to-one} because
        there are two possible $x$-values that can produce each given
        $y$-value.]
        {\
          \begin{tikzpicture}
            \begin{axis}[
                ylabel={$f(x)=x^2$},
                axis x line=bottom,
                axis y line=center,
                tick align=outside,
                yticklabels={,,}
                xticklabels={,,}
                xtickmax=10,
              ]
              \addplot[smooth,red]{x^2};
            \end{axis}
          \end{tikzpicture}
        }
        \hspace{0.2in}%
        \subfigure[The function $f(x)=x^3$ is \emph{one-to-one} because every
        given $y$-value is mapped from a unique $x$-value.]
        {\
          \begin{tikzpicture}
            \begin{axis}[
                ylabel={$f(x)=x^3$},
                axis x line=bottom,
                axis y line=center,
                tick align=outside,
                yticklabels={,,}
                xticklabels={,,}
                xtickmax=10,
              ]
              \addplot[smooth,blue]{x^3};
            \end{axis}
          \end{tikzpicture}
        }
    \end{center}
  \end{figure}
  A function $y=f(x)$ is one-to-one iff its graph intersects each horizontal
  line at most once.\index{horizontal line test}

  \index{onto}
  \index{surjective}
  $f: D \to R $ is \textbf{surjective} or \textbf{onto} iff
    \[\forall (y \in R) \exists  (x \in D) \big[f(x)=y\big]. \]

\begin{figure}[H]
    \begin{center}
        \subfigure[The function $f(x)=x^2$ is not \emph{surjective} because
        the values $(-\infty, 0)$ are never reached in its range.]
        {\
          \begin{tikzpicture}
            \begin{axis}[
                ylabel={$f(x)=x^2$},
                axis x line=bottom,
                axis y line=center,
                tick align=outside,
                yticklabels={,,}
                xticklabels={,,}
                xtickmax=10,
              ]
              \addplot[smooth,red]{x^2};
            \end{axis}
          \end{tikzpicture}
        }
        \hspace{0.2in}%
        \subfigure[The function $f(x)=x^3$ is \emph{one-to-one} because all $y$ values from $-\infty, \infty)$ have corresponding $x$-values.]
        {\
          \begin{tikzpicture}
            \begin{axis}[
                ylabel={$f(x)=x^3$},
                axis x line=bottom,
                axis y line=center,
                tick align=outside,
                yticklabels={,,}
                xticklabels={,,}
                xtickmax=10,
              ]
              \addplot[smooth,blue]{x^3};
            \end{axis}
          \end{tikzpicture}
        }
    \end{center}
  \end{figure}

    \index{bijective}
    A function $f:A \to B$ is \textbf{bijective} iff it is \emph{both injective and surjective}.

\begin{figure}[H]
    \begin{center}
        \subfigure[The function $f(x)=x^2$ is not bijective.]
        {\
          \begin{tikzpicture}
            \begin{axis}[
                ylabel={$f(x)=x^2$},
                xlabel={$x$},
                axis x line=bottom,
                axis y line=center,
                tick align=outside,
                yticklabels={,,}
                xticklabels={,,}
                xtickmax=10,
              ]
              \addplot[smooth,red]{x^2};
            \end{axis}
          \end{tikzpicture}
        }
        \hspace{0.2in}%
        \subfigure[The function $f(x)=x^3$ is bijective.]
        {\
          \begin{tikzpicture}
            \begin{axis}[
                ylabel={$f(x)=x^3$},
                xlabel={$x$},
                axis x line=bottom,
                axis y line=center,
                tick align=outside,
                yticklabels={,,}
                xticklabels={,,}
                xtickmax=10,
              ]
              \addplot[smooth,blue]{x^3};
            \end{axis}
          \end{tikzpicture}
        }
    \end{center}
  \end{figure}

\subsection{Graphs} \index{graphs}

  \index{graph}
    If $f$ is a function with a domain $D$, then its \textbf{graph} is the set
    \[ \Big\{ \big( x,f(x) \big) \Big | x \in D \Big\},\]
    that is, it is the set of all points $(x, f(x))$ where $x$ is in the domain of the function.%
\footnote{Here, the difference between the words \emph{graph} and \emph{plot} is sometimes confusing. Technically speaking, a \emph{graph} is the set defined explicitly here, while a function's \emph{plot} refers to any pictorial representation of a data set. However, since the usage is inconsistent in this text, these formal definitions will usually not apply. It can be safely assumed that as long as we are within the realm of real numbers, all uses of either \emph{graph} or \emph{plot} hereafter simply refer to the pictorial representation of a function's graph in the form of a curve on the cartesian plane.}

If $ (x,y) $ is a point on $f$, then $y=f(x)$ is the height of the graph above point $x$.
This height might be positive or negative, depending on the sign of $f(x)$.
We use this height relationship to plot functions.
\begin{figure}[H]
    \begin{center}
        \begin{tikzpicture}
          \begin{axis}[
              ylabel={$f(x)$},
              xlabel={$x$},
              axis x line=bottom,
              axis y line=center,
              tick align=outside,
              yticklabels={,,}
              xticklabels={,,}
              xtickmax=10,
            ]
            \addplot[smooth,red]{x+2};
          \end{axis}
        \end{tikzpicture}
      \caption{A plot of the function $f(x)=x+2$}
    \end{center}
  \end{figure}