What do the compositions of functions represent?

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Homework Help Overview

The discussion revolves around the compositions of a function representing the amount of money accumulated from an initial investment at a fixed interest rate. The specific function A(x) = 1.04x models the growth of an investment over time with annual compounding interest.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the process of composing the function A multiple times and question what these compositions signify in the context of investment growth over several years. There is also discussion about deriving a general formula for n compositions of A.

Discussion Status

The conversation is progressing with some participants confirming their understanding of the compositions and their implications. Hints have been provided to guide the original poster towards recognizing the pattern in the compositions, but a formal formula for n copies of A has yet to be established.

Contextual Notes

Participants are working within the constraints of a homework assignment, which may limit the information they can use or the methods they can apply. There is an emphasis on understanding the representation of the compositions rather than simply calculating them.

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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!
 
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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?
 
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?
 
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?
 

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