The function that best shows the relationship between n and f(n) is: f(n) = 48 * (0.5) ^ (n - 1).
Let's analyze the given information. The data pairs are (1, 48), (2, 24), and (3, 12). The pattern indicates that the number of alligators is decreasing by a factor of 0.5 each week.
The general form of exponential decay is given by f(n) = a * (r)^(n - 1), where "a" is the initial value, "r" is the common ratio, and "n" is the week number.
Comparing the given options:
f(n) = 48 * (0.5) ^ (n - 1): This function correctly represents the given data. It starts with an initial value of 48 and decreases by a factor of 0.5 each week.
f(n) = 48 * (0.5) ^ n: This function starts with an initial value of 48, but it doesn't match the given data as it would produce 48, 24, 12 for n = 1, 2, 3.
f(n) = 24 * (0.5) ^ n: This function starts with an initial value of 24, and the common ratio is 0.5, but it doesn't match the given data.
f(n) = 96 * (0.5) ^ (n - 1): This function starts with an initial value of 96, and the common ratio is 0.5, but it doesn't match the given data.
Therefore, the correct function that best shows the relationship between n and f(n) is f(n) = 48 * (0.5) ^ (n - 1).