This example presents a simple (non-core) elementary function, $o(x)$, defined by combining two core functions, $f$ and $g$ with a choice of arithmetic operations or composition. The function is visualized with a mapping diagram and a graph.

Notice how the arrows on the mapping diagram are paired with the
point on the graph of the function $o(x)$.

You can move the point for $x$ on the mapping diagram to see how
the function value changes both on the diagram and on the graph.

Sliders: The slider labeled comb_{op} controls whether an arithmetic operation or composition is applied to $f $ and $g$.

For selection of arithmetic operation: 1.

For composition: 2 or 3.

The slider labeled bin_{op} appears when comb_{op} = 1. It controls which arithmetic operation is applied to $f $ and $g$ as $f$ bin_{op} $g$.

Notice the apparent discontinuity of the function that uses division $o(x) = f(x)/g(x)$ on the mapping diagram is
evidenced by the missing arrow as the value of $g(x)$
passes through $0$ when $x=0, \pm \pi,$ and $\pm 2\pi$.

Check the box labeled Show/Hide Table Data to display a selection of
data on the graph with points on the graph and
with arrows on the diagram to match the data in a table.

The functions $f$ and $g$ can be changed by making entries in the appropriately labeled input boxes.

Martin Flashman, March. 14, 2017. Created with GeoGebra