# A ∘ A , A ∘ A ∘ A a n d A ∘ A ∘ A ∘ A of the Function A ( x ) = x + 0.05 x = 1.05 x . To find a formula for n copies of A.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.6, Problem 67E
To determine

## To Find: A∘A, A∘A∘A and A∘A∘A∘A of the Function A(x)=x+0.05x=1.05x . To find a formula for n copies of A.

Expert Solution

The answer is AA=(1.05)2x, AAA=(1.05)3x and AAAA=(1.05)4x. And, for n copies of A the formula becomes =(1.05)nx .

### Explanation of Solution

Given information : The function A(x)=x+0.05x=1.05x is given.

Formula used : The concept of Combining Functions is used.

Calculation :

The functions of compound interest is given A(x)=x+0.05x=1.05x

AA=A(A(x))AA=A(1.05x) AA= (1.05) 2 xAA=1.1025x

AAA=A(A(A(x)))AAA=A(A(1.05x)) AAA=A( (1.05) 2 x) AAA= (1.05) 3 xAAA=1.157625x

AAAA=A(A(A(A(x))))AAAA=A(A(A(1.05x))) AAAA=A(A( (1.05) 2 x)) AAAA=A( (1.05) 3 x) AAAA= (1.05) 4 xAAAA=1.21550625x

These Compositions represent the compound interest on previous year’s amount.

For n copies of A the formula becomes =(1.05)nx

Hence, the answer is AA=(1.05)2x, AAA=(1.05)3x and AAAA=(1.05)4x. And, for n copies of A the formula becomes =(1.05)nx .

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