Discussion
Consider the following functions for non-zero positive integers, \(p\) and \(q\).
\[ \begin{aligned} f(p,q) &= \underbrace{p \times p \times p \times \cdots \times p}_{q\ \text{terms}} \\ &= p^{q}; \quad f(p,1) = p \end{aligned} \]
\[ \begin{aligned} g(p,q) &= p^{p^{p^{p^{\cdot^{\cdot^{\cdot}}}}}}\ (\text{up to } q \text{ terms}); \\ g(p,1) &= p \end{aligned} \]
Which one of the following options is correct based on the above?
Answer: A
The correct option is f(2,2) = g(2,2).
f(p,q) is p raised to the power q, and g(p,q) is a tower of q copies of p. With two terms each:
\[ \begin{aligned} f(2,2) &= 2^{2} = 4 \\ g(2,2) &= 2^{2} = 4 \end{aligned} \]
The other options are false. f(g(2,2),2) = f(4,2) = 16 and f(2,g(2,2)) = f(2,4) = 16, so neither is less than the other. g(2,1) = 2 = f(2,1) by definition, so they are equal. f(3,2) = 9 while g(3,2) = \(3^{3}\) = 27, so f(3,2) is smaller, not greater.
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