The goal of this discussion is for students to articulate an appropriate domain for given the context. Begin the discussion by selecting students to share their definitions and recording these for all to see. If not brought up by students, connect the definition for the term of with a general form of an exponential equation, such as , where represents the starting term and represents the growth factor. Students should understand that, since is a geometric sequence, it is a type of exponential function that has a domain restricted to a subset of the integers.

Specifically, the domain of without regard to the context is the integers where . However, realistically, there will eventually be too little water to reasonably pour out of it. can be quickly evaluated using technology, for example, and the table in this activity can be extended to show successive inputs. Deciding the amount of water left that is too small to reasonably dispense is up to the modeler. After 12 people fill their bottles, there is less than 1% of the original amount of water left (about 10 ounces). After 18 people, there is less than 0.1% of the original amount of water left, or about 0.9 ounce. After 20 people, there is only 0.4 ounce of water left in the jug.

Some students may argue that the amount of water is still measurable, so a reasonable domain is higher. Others may argue that no one is going to try and add 0.4 ounces to their water bottle, so an input of 20 should be the highest value in the domain. Let students know that both of these answers could be correct depending on what the modeler is trying to represent with the function they create.

Regarding notation, it is not crucial at this time that students represent the domain using inequality notation. For example, for the last question, they might say “integers from 0 to 20.”

Some students may not realize that it is okay to use both and when defining a function recursively. Encourage these students to think about how the “previous side lengths” for term  can be expressed using function notation.

Use this discussion to make sure students understand that sometimes recursive forms are a good choice over the more familiar term forms.

Display the sequence 1, 1, 2, 3, 5, 8, 13 and the recursive definition , for all to see and check their answer against. Either give students a minute to think about how to write a definition for the term of the sequence, or just tell them that it is more complicated to derive than class time allows. An important modeling point is that the goal is to create a model that suits the needs of what we are trying to do. If we wanted to know the value of , then a spreadsheet that adds the previous two terms to get the next term, which is a recursive definition, is going to be faster than working out a definition for the term.

Ask students, "What is the domain of this function? Does it have any restrictions?" (The integers where or equivalent. We can always add another square onto the image, even if only in our imaginations once the squares get too big.) Select students to share their responses.

If time allows, tell students that we could define the Fibonacci sequence as , and we would still get the same list of numbers.