Example QF.COMP.2

Example QF.COMP.2 :$ f(x) = x - 2; g(x) = x^2$
The composition of the core quadratic function $g(x)=x^2$ followed by the core linear function $f(x) = x-2$, so $q(x) = f(g(x)) = x^2 - 2$.
Draw a mapping diagram yourself or use the diagram created with GeoGebra to explore the diagram further.
Compare the mapping diagram with the graphs of $g(x)$ and $f(x)$

Graphs of $g(x)$ and $f(x)$

Mapping Diagram Showing Composition.

Given a point / number, $x$, on the source line, there is a blue arrow meeting the target line at the point / number, $ x^2 - 2$
This point corresponds to the quadratic function's value for $x$.
The values for the core mapping diagram for $x-2$ in red are applied to the values $g(x)=x^2$ (green) .

As $x$ increases, $q(x)$ decreases to value $q(0)=-2$ and then increases.
For any $a \gt 0$ the even symmetry with respect to $x=0$ of $q$ gives $q(+a) = q(-a)$