Example QF.2.3 : $A =\frac 1 2; q(x) = \frac 1 2 x^2$
This is best understood as the composition of the core quadratic function $g(x)= x^2$ followed by the linear function $f(x) = \frac 1 2 x$, so $q(x) = f(g(x))$.
Draw a mapping diagram showing this composition  or have SAGE create this by evaluating.
Compare the mapping diagram with the graphs of $g(x)$ and $q(x)$
 Graphs of $g(x)$ and $q(x)$ 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, $\frac 1 2 x^2$. This point corresponds to the quadratic function's value for $x$. he values for the core mapping diagram for $x^2$ in green are magnified by a factor of $\frac 1 2$  shown by the red arrows.

As $x$ increases, $q(x)$ decreases to value $q(0)=0$ and then increases.