You have the following information:
- Length x of the square plate can be no more than 12 inches.
- Area y of the square plate can be at most 144 inches.
In order to determine which is the domain and the range. You consider the given restriction respect to length and area:
x ≤ 12 length can be no more than 12 in
y ≤ 144 area must be at most 144 in
The previous inequalities determines the domain and range. Due to the area depends on the length of the square, the domain of the function are the available values of the length. And then, the domain is also determined by the values of y which are the result of such a domain. Thus, you have:
domain: 0 < x ≤ 12
range: 0 < y ≤ 144
Both intervals start with 0 because it is not negative lengths or areas.