Analogical Reasoning Math Example 3

Solution

1. 1
   Analogy: repeated multiplication of $a$ by itself $n$ times gives $a^n$. Repeated application of $f$ to itself $n$ times gives $f^n = f \circ f \circ \cdots \circ f$ ($n$ times).
2. 2
   So $f^2 = f \circ f$ (apply $f$ twice) and $f^m \circ f^n = f^{m+n}$ (apply $f$ a total of $m+n$ times).
3. 3
   Caution: $f^2(x) \ne [f(x)]^2$ in general — the analogy is with composition, not multiplication.