What really are functions added together? I saw an example of two separate events happening distance per time. 1) a train was accelerating at $ t^2 $ i.e., $ f(t) = t^2 $ and, 2) inside the train a man was moving forward at $ .01t $, i.e., $ f(t) = .01t $ and the equation given was then
$ f(t) = t^2 + .01t $
Can I think of this as function addition, i.e. adding two separate functions together -- like in all the beginning math textbooks when they show composition? e.g.:
$ f(t) = 3t^2 - 4 $ and $ g(t) = 4t + 5 $
$ \Rightarrow $
$ (f + g)(t) = 3t^2 +4t + 1 $
It seems like any polynomial $ f(x) = c_1x^n + c_2x^{n-1} + \dots c_nx  $ can be taken apart into individual functions. True? I've only seen simple things like this train example and typical stuff like figuring profit when you sell at $ 9.5x $ but have to consider costs like $ - 2x - 1500_{startup} $ for the final calculation $ f(x) = 7.5x - 1500_{startup}$.
But in the train example, just looking at $ f(t) = t^2 + .01t $ doesn't tell me that the $ .01t $ term is the deciding, germane thing. From the description I know that we're dealing with the man's total speed, i.e., the train's plus his speed. But again, it's not so obvious, other than the obvious fact that we're adding up components.
Now, how would you express a situation where the man is walking forward in the train and throwing a ball forward? What is the ball's total speed? I'm guessing Galileo's parabola equation would have to be included here -- specifically for the horizontal component. Not sure how, though. In general, can anyone point me to non-trivial examples of function compositing?
 A: Here's how I teach the equivalent in my 12th grade Physics class. This applies to vectors of any numbers of dimension, not just one dimension as in your examples.
Let $s_{ab}$ be the position vector of object $a$ relative to object $b$, $s_{bc}$ be the position vector of object $b$ relative to object $c$, and so on. Then we get these equations for comparing frames of reference:
$$s_{ab}=-s_{ba}$$
$$s_{ab}+s_{bc}=s_{ac}$$
Taking derivatives with respect to time, we get similar equations for velocities and accelerations.
So in the example of your train $t$, man $m$, ball $b$, and earth $e$, we get
$$s_{be}=s_{bm}+s_{mt}+s_{te}$$
I.e. the position of the ball relative to the earth equals the balls position relative to the man plus the man's position relative to the train plus the train's position relative to the earth. The same sum applies to their velocities.
Any position, velocity, or acceleration is relative to some frame of reference. Usually it is obvious which frames are meant, but not always. These equations help to keep things straight.
Does that answer some of your questions?
