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.

Definition: FC Function Composition.

Two linear examples.
In these two examples we use GeoGebra to visualize composition for linear functions.
Example LF.COMP.1 : $f(x) = 2x -1; g(x) = 3x + 2$
Example LF.COMP.2 $g(x) = 2x - 1; f(x) = 3x + 2$

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.

Example LF.COMP.3 Suppose $g(x) =3x$ and  $f(x)=x+2$ then $(f \circ g)(x) = 3x + 2;$

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