Final answer:
The reasonable domain values for the function f(x) = 1.25x^2 are 0 ≤ x ≤ 10, and the reasonable range values are 0 ≤ f(x) ≤ ∞.
Step-by-step explanation:
The function f(x) = 1.25x^2 models the packaging costs for a box shaped like a rectangular prism. The side lengths of the box are given as 2x in., 2x in., and 0.5x in. We are asked to find the reasonable domain and range values for this function, considering that the longest side length of the box can be no greater than 20 in.
The domain of the function represents the possible input values (x-values) for the function. In this case, x represents the side length of the box and it cannot exceed 20 in., so the reasonable domain values are 0 ≤ x ≤ 10.
The range of the function represents the possible output values (y-values) for the function. Since squaring a positive number always yields a positive result, the function f(x) = 1.25x^2 will always produce non-negative values for any input. Therefore, the reasonable range values for this function are 0 ≤ f(x) ≤ ∞, where ∞ represents positive infinity.