Question

If you invest x dollars at 4% interest compounded annually, then the amount A(x) of the investment after one year is \(A\left( x \right) = {\bf{1}}.{\bf{04}}x\). Find \(A \circ A\), \(A \circ A \circ A\), and \(A \circ A \circ A \circ A\). What do these compositions represent? Find a formula for the composition of n copies of A.

Short answer

The functions are \(A \circ A\left( x \right) = {\left( {1.04} \right)^2}x\), \(A \circ A \circ A\left( x \right) = {\left( {1.04} \right)^3}x\), and \(A \circ A \circ A \circ A\left( x \right) = {\left( {1.04} \right)^4}x\).

The functions \(A \circ A\), \(A \circ A \circ A\), and \(A \circ A \circ A \circ A\) represents the amount of investment after 2, 3, and 4 years. For n copies of A the formula \(A \circ A \circ \cdot \cdot \cdot \circ A\left( x \right) = {\left( {1.04} \right)^n}x\).

Step-by-step solution

Find the composition \(A \circ A\)

Thecomposite function\(A \circ A\) can be expressed as:

\(\begin{aligned}A \circ A\left( x \right) &= A\left( {A\left( x \right)} \right)\\ &= A\left( {1.04x} \right)\\ &= 1.04\left( {1.04x} \right)\\ &= {\left( {1.04} \right)^2}x\end{aligned}\)

Find the composition \(A \circ A \circ A\)

The composite function \(A^\circ A^\circ A\) can be expressed as:

\(\begin{aligned}A^\circ A^\circ A\left( x \right) &= A\left( {A^\circ A\left( x \right)} \right)\\ &= A\left( {{{\left( {1.04} \right)}^2}x} \right)\\ &= 1.04\left( {{{\left( {1.04} \right)}^2}x} \right)\\ &= {\left( {1.04} \right)^3}x\end{aligned}\)

Find the composition \(A \circ A \circ A \circ A\)

The composite function \(A \circ A \circ A \circ A\) can be expressed as:

\(\begin{aligned}A \circ A \circ A \circ A\left( x \right) &= A\left( {A \circ A \circ A\left( x \right)} \right)\\ &= A\left( {{{\left( {1.04} \right)}^3}x} \right)\\ &= 1.04\left( {{{\left( {1.04} \right)}^3}x} \right)\\ &= {\left( {1.04} \right)^4}x\end{aligned}\)

Find the composition \(A \circ A \circ  \cdot  \cdot   \cdot  \circ A\)

Based on the given patterns the function \(A \circ A \circ \cdot \cdot \cdot \circ A\) composite function can be expressed as:

\(A \circ A \circ \cdot \cdot \cdot \circ A\left( x \right) = {\left( {1.04} \right)^n}x\)

So, the functions \(A \circ A\), \(A \circ A \circ A\), and \(A \circ A \circ A \circ A\) represents the amount of investment after 2, 3, and 4 years. For n copies of A the formula \(A \circ A \circ \cdot \cdot \cdot \cdot \cdot \cdot \circ A\left( x \right) = {\left( {1.04} \right)^n}x\).