Consider two linear function $f(x) = m_fx + b_f $ and $g(x) = m_gx +
There is a very important way to combine these functions that is
fundamental to understanding linear functions and a vast number of
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)
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
Two linear examples.
In these two examples we use GeoGebra to visualize composition for
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
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 $g$.
LF.DCOMP.0 Dynamic Visualization of Composition for
Linear Functions: Graphs, and Mapping Diagrams