Consider two linear function $f(x) = m_fx + b_f $ and $g(x) = m_gx +
b_g $.
There is a very important way to combine these functions that is
fundamental to understanding linear functions and a vast number of
other functions:
First find $g(x)$, then find $f(g(x))$.
We can see this as a new function called the composition of $f$
with $g$ , denoted usually as $f \circ g$.
So $(f \circ g)(x) = f(g(x))$ and $f \circ g : x \rightarrow g(x)
\rightarrow f(g(x))$.
In this section we will study how mapping diagrams visualize and
illuminate some fundamental features of linear functions by
understanding composition. Though a visualization of composition is
possible with graphs, the simplicity of the mapping diagram for
composition makes it the choice for visualization as a learning
tool. But first we will formally define the key concept of function
composition.
Two linear examples.
In these two examples we use GeoGebra to visualize composition for
linear functions.
Compositions are a key to understanding linear functions. the key
result is not hard to understand, but is very useful.
This result says linear functions are built by composing two
core linear functions: constant addition and
scalar multiplication.
Here is an example that visualizes the theorem.
You can use this next dynamic example
with GeoGebra to investigate further the effects of composition in a
mapping diagram of $f$ composed with a mapping diagram of $g$.
Example
LF.DCOMP.0 Dynamic Visualization of Composition for
Linear Functions: Graphs, and Mapping Diagrams