   Chapter 1.3, Problem 62E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

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

To determine

To find: The composition of AAAA and interpret it; also find an explicit formula for the composition of n copies of A.

Explanation

Given:

It is given that x dollars invested at 4% interest compounded annually. Then the invested amount after one year is A(x)=1.04x .

Calculation:

The composite function (fg)(x) is defined as, (fg)(x)=f(g(x)) .

First, find the composition of (AA)(x)=A(A(x)) .

Substitute A(x)=1.04x for x in A(x) ,

(AA)(1.04x)=1.04(1.04x)=(1.04)2(x)

Therefore, the composition of (AA)(x)=(1.04)2(x) . It represents the amount after two years.

Similarly to find (AAA)(x)=A((AA)(x)) substitute (AA)(x)=(1.04)2x for x in A(x) ,

(AAA)(x)=A((AA)(x))=1.04(1

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