Chapter 2.6, Problem 67E

To determine

To Find: $A\circ A,A\circ A\circ AandA\circ A\circ A\circ A$ of the Function $A(x)=x+0.05x=1.05x$ . To find a formula for n copies of A.

Answer to Problem 67E

The answer is $A\circ A={(1.05)}^{2}x,A\circ A\circ A={(1.05)}^{3}x\text{ and }A\circ A\circ A\circ A={(1.05)}^{4}x.$ And, for n copies of A the formula becomes $={(1.05)}^{n}x$ .

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 $A(x)=x+0.05x=1.05x$

  $\begin{array}{l}A\circ A=A(A(x))\\ \Rightarrow A\circ A=A(1.05x)\\ $ \Rightarrow A\circ A={(1.05)}^{2}x$\Rightarrow A\circ A=1.1025x\end{array}$

  $\begin{array}{l}A\circ A\circ A=A(A(A(x)))\\ \Rightarrow A\circ A\circ A=A(A(1.05x))\\ $ \Rightarrow A\circ A\circ A=A({(1.05)}^{2}x)$$ \Rightarrow A\circ A\circ A={(1.05)}^{3}x$\Rightarrow A\circ A\circ A=1.157625x\end{array}$

  $\begin{array}{l}A\circ A\circ A\circ A=A(A(A(A(x))))\\ \Rightarrow A\circ A\circ A\circ A=A(A(A(1.05x)))\\ $ \Rightarrow A\circ A\circ A\circ A=A(A({(1.05)}^{2}x))$$ \Rightarrow A\circ A\circ A\circ A=A({(1.05)}^{3}x)$$ \Rightarrow A\circ A\circ A\circ A={(1.05)}^{4}x$\Rightarrow A\circ A\circ A\circ A=1.21550625x\end{array}$

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

For n copies of A the formula becomes $={(1.05)}^{n}x$

Hence, the answer is $A\circ A={(1.05)}^{2}x,A\circ A\circ A={(1.05)}^{3}x\text{ and }A\circ A\circ A\circ A={(1.05)}^{4}x.$ And, for n copies of A the formula becomes $={(1.05)}^{n}x$ .