Solution: This is a comprehensive problem: examining parity, periodicity, piecewise function, composite function, the relationship between the solution of equation and function, and the idea of combining numbers with shapes. This is a well-chosen ending.

According to the topic: f(x) is a odd function with a period of 2, and the image of f(x) can be drawn through the combination of numbers and shapes, from which the analytical formula of f(x) in a period can be written, and F (x) =-2x-2- 1

F2(x)=f(f(x))=f(2x) or f (-2x) = = > F3(X)=f(4x) or f(-4x).

Therefore, let t=4x, the original equation is decomposed into f(t)=4.5/(t-4), and f(t)= -4.5/(t-4) t belongs to [-4, 12].

The sum of all solutions of the original equation = (t 1+t2+...+tn)/4.

Draw images of functions f(t), 4.5/(t-4) and -4.5/(t-4) by combining numbers and shapes (please draw a picture yourself). The equation f (t) = 4.5/(t-4) * * has three sets of solutions.

Equation f(t)=-4.5/(t-4)*** has four groups of solutions; These seven groups of solutions are all symmetric about the center of point (4,0), so the sum of each group of solutions is 8.

Therefore, the sum of all solutions of the original equation = (t1+T2+...+TN)/4 = (7 * 8)/4 =14. = = > choose C.