Consider the following functions for non-zero positive integers, p and q.

$\mathrm{f}(\mathrm{p},\mathrm{q})=\frac{\mathrm{p}\times \mathrm{p}\times \mathrm{p}\times .......\times \mathrm{p}}{\mathrm{q}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{s}}={\mathrm{p}}^{\mathrm{q}};$f(p, 1) = p

$g(p,q)=p^{p^{p^{p^{p^{.{.}^{.{.}^{.{.}^{uptoqterms}}}}}}}}$; g(p, 1) = p

Which one of the following options is correct based on the above?

The correct option is 𝑓(2,2) = 𝑔(2,2).

Let us analyze the given functions and evaluate their values step-by-step to show why this option is correct.

1. Understanding the function 𝑓(p, q):
The function 𝑓(p, q) is defined for positive integers as:
$$f(p,q)=p^q$$

Let's calculate the value of 𝑓(2, 2) by substituting p = 2 and q = 2:
$$f(2,2)=2^2=4$$

2. Understanding the function 𝑔(p, q):
The function 𝑔(p, q) represents tetration, which is repeated exponentiation of p up to q terms (associated from top to bottom, or right to left):
$$g(p,q)=p^{p^{p^{\cdots }}}\text{up to }q\text{ terms}$$ With the base case:
$$g(p,1)=p$$ For q = 2, we have exactly 2 terms in the power tower:
$$g(p,2)=p^p$$

Let's calculate the value of 𝑔(2, 2) by substituting p = 2 and q = 2:
$$g(2,2)=2^2=4$$

3. Comparing the two values:
Since we have:
𝑓(2, 2) = 4
𝑔(2, 2) = 4
Therefore, we can conclude that:
$$f(2,2)=g(2,2)$$ This confirms that the first option is indeed correct.