What do the compositions of functions represent?

[metalmagik]

Homework Statement

If you invest x dollars at 4% interest compounded annually, then the amount A(x) = 1.04x. Find A o A, A o A o A, and A o A o A o A. What do these compositions represent? Find a formula for the composition of n copies of A.

Homework Equations

f o g = f(g(x))

The Attempt at a Solution

Okay, well, I'm trying to work this out and I get really stuck.
If I do A(A(x)), that's A(A(x)) = 1.04(1.04x) correct? I'm not sure at all "what these compositions are supposed to represent. Please help!

 

[nicksauce]

Correct for A(A(x)).

Hint: If I have x dollars to start with, how many do I have after year 1? After year 2 and 3?

 

[metalmagik]

OH okay. I think I get it. So when they ask to find each composition, I just put A(A(x)), then A(A(A(x))) and so on?

I get that each composition represents the TOTAL amount that is invested up to year 2, 3, 4 and so on, but how do I find the formula for n copies of A?

 

[HallsofIvy]

Just as you say, A o A= (1.04)(1.04x)= (1.04)²x. So A o A o A= (1.04)(1.04)²x= (1.04)³x. A o A o A o A= (1.04)(1.04)³x= (1.04)⁴x. Now, suppose you had A o A o A o A o A o A o A o A, with eight A's. What would that be?