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\centerline{ \begin{Large}The Definition of a Function of a Real Number\end{Large} }
\vspace{5mm}
Let $A$ and $B$ be sets. If $f$ is a relation from $A$ to $B$ then $f$ is called a $\textit{function}$ from $A$ to $B$ if for every $a \in A$ there exists a unique $b \in B$ such that $\left( a, b \right) \in f$.
Or equivalently,
If $ A $ and $ B $ are sets, then a $\textit{function}$, $f$, from $A$ to $B$ is the set of ordered pairs $\left( a, b \right)$ contained in $A \times B$ such that if $\left( a, b \right) \in f $ and $ \left( a, b^{ \prime } \right) \in f$ then $b = b^{ \prime }$.
Please take note of the following:
\begin{itemize}
\item The notation $f : A \rightarrow B$ is used to demonstrate that $f$ $\textit{maps}$ $A$ into $B$.
\item Set $A$ is called the $\textit{domain}$ of the function and usually notated $D \left( f \right) $.
\item The image of $a$ under $f$ is notated $f \left( a \right) = b$ and all elements of this set is called the range and can be notated $R \left( f \right) = \{ f \left( x \right) \ | \ x \in D \left( f \right) \}$.
\item The range is a subset of $B$.
\end{itemize}
Questions:
\begin{enumerate}
\item The second formulation of the definition of a function of a real number makes use of the Cartesian or set product. How is this product defined?
\item What subset of the Cartesian plane is defined under $A \times B$ if:
\begin{enumerate}
\item $A = \{1, 2, 3, 4 \}$ and $B = \{0, 1, 2, 3, 4 \}$
\item $A = \{1, 2, 3 \}$ and $B = \mathbb{R}$
\item $A = \left[ -2, 3 \right]$ and $B = \left[5, 8 \right]$
\item $A = \mathbb{R}$ and $B = \{-3, -1, 1, 3, 5 \}$
\end{enumerate}
\item Let $f$ be the squaring function such that $f : A \rightarrow \mathbb{R}$. What is the range or the image of $A$ under $f$ if $A =$
\begin{enumerate}
\item $\{ -5, -3, -1, 0, 2, 4, 6, 8\}$
\item $\mathbb{R}$
\item $\{ 2, 3 \} \cup \left[ 6, 11 \right]$
\end{enumerate}
\item When mapping the function $f$ in the Cartesian plane does the graph "appear" any different than $A \times \mathbb{R}$?
\item Which of the domains, if any, in question 3 result in an onto function? Which of the domains, if any, result in a function that is one-to-one?
\item Can a function be onto and not one-to-one or be one-to-one and not onto? What is the significance of these properties?
\end{enumerate}
\vspace{10mm}
$\textbf{Definitions Frequently Used in Association with Functions}$
If $a, b \in \mathbb{R}$ and $a < b $ then $\left( a, b \right) = \{ x \in \mathbb{R} \ | \ a < x < b \}$ is called an $\textit{open interval}$, and $a$ and $b$ are called $\textit{endpoints}$.
If $a, b \in \mathbb{R}$ and $a < b$ then $\left[ a, b \right] = \{ x \in \mathbb{R} \ | \ a \le x \le b \}$ is called a $\textit{closed interval}$, and $a$ and $b$ are called $\textit{endpoints}$.
A function $f : A \rightarrow B$ is called $\textit{onto}$ or $\textit{surjective}$ if and only if for every $y \in B$ there exists an $x \in A$ such that $y = f \left( x \right)$.
A function $f : A \rightarrow B$ is called $\textit{one-to-one}$ or $\textit{injective}$ if and only if for every $x_1, x_2 \in A$ if $f \left( x_1 \right) = f \left( x_2 \right)$ then $x_1 = x_2$
Geometrically speaking, a function $f : A \rightarrow B$ is surjective if it passes the first horizontal line test, namely, if $b \in B$ then the horizontal line $y = b$ intersects $f$ at least once. $f$ is injective if it passes the second horizontal line test, namely, the horizontal line $y = b$ intersects $f$ at most once.
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