functions within functions

I am wondering how one uses a function within a function.  For exampe:

t=0,1..10

f(t):=2*t

then t and f(t) are vectors of length 10.  t contains the 11 numbers 0-10 and f(t) contains the 11 numbers 0 through 20

Now, I need to use the values of f(t) in another equation, like:

g(f(t)):=10*f(t)

I should be able to get a vector of 11 numbers ranging from 0 to 200, but I can not make this work. 

In my actual  problem I am using an ODE solver to get the values f(t) and so I can't just change the algebra.  I need to use the values of f(t) in another equation?  HOW DO I DO THIS