**AEF.AOEF Arithmetic Operations and Elementary Functions
**

The elementary functions are defined initially from a list of core functions.

In this section we focus on visualizing the use of arithmetic operations with mapping diagrams to make sense of elementary functions formed by using arithmetic.

and two secondary binary arithmetic operations, subtraction $-$ and division $\div$.

The algebraic properties of these operations are studied from early grade school, and tables are frequently used to present the common arithmetic facts. No visualization is attached to operations that connects them to graphs except much later in school when graphs of functions of two variables are introduced.

Mapping diagrams provide a simple and consistent tool for making sense of binary operations without requiring a more subtle understanding required for graphs.

One can consider a binary operation as a function of
two variables, with the value of the function being the result of
applying the operation to an ordered pair of numbers.

To visualize the operation (function) we use two (ordered) vertical axes for the ordered pair of numbers to which the operation (function) is applied, and a third vertical axis where the result is marked.

From a single number on the first and second axes an arrow
emanates meeting together at the number on the third axis corresponding
to the result of the operation.To visualize the operation (function) we use two (ordered) vertical axes for the ordered pair of numbers to which the operation (function) is applied, and a third vertical axis where the result is marked.

GeoGebra Mapping Diagram for Arithmetic Operations

$n=1: +, n=2: -,n=3: \times, n=4 : \div$

Change $n$ with the slider.

We apply an arithmetic operation, $O$, to functions,
$f_1$ and $f_2$, to create a new function, $f$. Let $f(x) =
O(f_1(x),f_2(x))$.

For example, if $O(a,b)=a+b$, then $f(x)=f_1(x)+f_2(x)$.

Visualizing this algebra with mapping diagrams requires only the addition of one more axis representing the domain of the function $f$ to the diagram for the arithmetic operation,

Each number, $x$, in the domain of $f$, is visualized by a point on the corresponding axis. Two arrows are drawn to points on separate axes for the values of $f_1(x)$ and $f_2(x)$. Arrows are drawn from these two points (values) to a point on the final axis representing the value of $f(x)=O(f_1(x),f_2(x))$.

Continuing the example, if $O(a,b)=a+b$, and $f_1(x) =\sin(x)$ and $f_2(x)= x$ then $f(x)=f_1(x)+f_2(x)= \sin(x) + x$,

The corresponding mapping diagram is shown below with a GeoGebra figure [$n=1$].

Change the functions by entering different functions for $f_1$ and $f_2$.

Change $n$ with the slider to see other operations applied to the functions.

$n=1: +, n=2: -,n=3: \times, n=4 : \div$

For example, if $O(a,b)=a+b$, then $f(x)=f_1(x)+f_2(x)$.

Visualizing this algebra with mapping diagrams requires only the addition of one more axis representing the domain of the function $f$ to the diagram for the arithmetic operation,

Each number, $x$, in the domain of $f$, is visualized by a point on the corresponding axis. Two arrows are drawn to points on separate axes for the values of $f_1(x)$ and $f_2(x)$. Arrows are drawn from these two points (values) to a point on the final axis representing the value of $f(x)=O(f_1(x),f_2(x))$.

Continuing the example, if $O(a,b)=a+b$, and $f_1(x) =\sin(x)$ and $f_2(x)= x$ then $f(x)=f_1(x)+f_2(x)= \sin(x) + x$,

The corresponding mapping diagram is shown below with a GeoGebra figure [$n=1$].

Change the functions by entering different functions for $f_1$ and $f_2$.

Change $n$ with the slider to see other operations applied to the functions.

$n=1: +, n=2: -,n=3: \times, n=4 : \div$

A slightly more efficient mapping diagram uses only three axes to visualize the arithmetic of functions.

A single axis works for the pair of domain numbers in the binary arithmetic operation and the same single axis works for values of both $f_1(x)$ and $f_2(x)$.

This condensed mapping diagram is illustrated in the following GeoGebra figure.[$n=1$] with the corresponding graphs in the second frame of the figure.

Change the functions by entering different functions for $f_1$ and $f_2$.

Change $n$ with the slider to see other operations applied to the functions.

$n=1: +, n=2: -,n=3: \times, n=4 : \div$