AEF.COMP Composition and Elementary Functions

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

The definition continues with a recursive formulation, building more elementary functions from previously defined elementary functions by using arithmetic operations, composition of functions, and inverting functions.
In this section we focus on visualizing the composition of functions with mapping diagrams to make sense of elementary functions formed by using composition.

Visualizing composition of functions with mapping diagrams is not a new concept. It was covered first in the section on linear functions, specifically in the subsection LF.COMP: Composition of Linear Functions. In fact, linear composition has been discussed in almost every section of this resource!

Recall the definition:
The composition of $f$ with $g$ , denoted usually as $f \circ g$ is defined by $f \circ g (x)= f(g(x))$ where $f \circ g : x \rightarrow g(x) \rightarrow f(g(x))$.

Definition: FC Function Composition.

Two examples.
In these two examples we use GeoGebra to visualize composition for nonlinear functions.
Example AEF.COMP.1: $ f(x) = \sin(x); g(x) = x^2; f\circ g(x) = \sin(x^2)$
Example AEF.COMP.2: $ f(x) = \sin(x); g(x) = x^2;  g \circ f (x) = \sin^2(x)$

You can use this next example with user defined functions for $f$ and $g$ to investigate further the effects of composition in a mapping diagram  of $f \circ g$  or $g \circ f$.

Example AEF.DCOMP.0 Dynamic Visualization of Composition for Functions: Graphs, and Mapping Diagrams