Example TRIG.SYM.3  :$A=2; B=1; trig(x) = \cos(x); C = D =0; f(x)=2cos(x).$
The composition is the core trigonometric function $trig(x)=\cos$ followed by the core linear function $f_{*2}(x) = 2x$, so $f(x) = 2\cos(x)$.
Draw a mapping diagram showing this composition or use the diagram created with GeoGebra to explore the diagram further.
Compare the mapping diagram with the graphs of $\cos(x)$ and $f(x)$

For any $a \gt 0$ the even symmetry with respect to $x=0$ of $f_C$ gives $f(+a) = f(-a)$