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Foxes and Rabbits
Task
Given below is a table that gives the populations of foxes and rabbits in a national park over a 12 month period. Note that each value of $t$ corresponds to the beginning of the month.
$t$, month  1  2  3  4  5  6  7  8  9  10  11  12 

$R$, number of rabbits  1000  750  567  500  567  750  1000  1250  1433  1500  1433  1250 
$F$, number of foxes  150  143  125  100  75  57  50  57  75  100  125  143 
 According to the data in the table, is $F$ a function of $R$? Is $R$ a function of $F?$ Explain.
 Are either $R$ or $F\,$ functions of $t$? Explain.
IM Commentary
There is a natural (and complicated!) predatorprey relationship between the fox and rabbit populations, since foxes thrive in the presence of rabbits, and rabbits thrive in the absence of foxes. However, this relationship, as shown in the given table of values, cannot possibly be used to present either population as a function of the other. This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations. Noteworthy is that since the data is a collection of inputoutput pairs, no verbal description of the function is given, so part of the task is processing what the "rule form" of the proposed functions would look like.
The predatorprey example of foxes and rabbits is picked up again in FIF Foxes and Rabbits 2 and 3 where students are asked to find trigonometric functions to model the two populations as functions of time.
This task could be used early on when functions are introduced. It illustrates examples of functions as well as relationships that are not functions. It could also be used as an assessment item.
This task is adapted from "Functions Modeling Change", Connally et al, Wiley 2007.
Solution

The key is understanding that a function is a rule that assigns to each input exactly one output, so we will test the relationships in question according to this criterion:
For the first part, that is, for $F$ to be a function of $R$, we think of $R$ as the input variable and $F$ as the output variable, and ask ourselves the following question: Is there a rule, satisfying the definition of a function, which inputs a given rabbit population and outputs the corresponding fox population. The answer is no: We can see from the data that when $R=1000$, we have one instance where $F = 150$, and another where $F = 50$. Since this means that a single input value corresponds to more than one output value, $F$ is not a function of $R$. In the language of the problem's context, this says that the fox population is not completely determined by the rabbit population; during two different months there are the same number of rabbits but different numbers of foxes.
Similarly, we can see that if we consider $F$ as our input and $R$ as our output, we have a case where $F = 100$ corresponds to both $R = 500$ and $R = 1500$, two different outputs for the same input. So $R$ is not a function of $F$: There are two different months which have the same number of foxes but two different numbers of rabbits.

Letting $t$, months, be the input, we can clearly see that there is exactly one output $R$ for each value of $t$. That is, the rule which assigns to a month $t$ the population of rabbits during that month fits our definition of a function, and so $R$ is a function of $t$. By the same reasoning $F$ is also a function of $t$. Again, in the context of the situation it makes sense that at any given point in time, there is a unique number of foxes and a unique number of rabbits in the park.
Foxes and Rabbits
Given below is a table that gives the populations of foxes and rabbits in a national park over a 12 month period. Note that each value of $t$ corresponds to the beginning of the month.
$t$, month  1  2  3  4  5  6  7  8  9  10  11  12 

$R$, number of rabbits  1000  750  567  500  567  750  1000  1250  1433  1500  1433  1250 
$F$, number of foxes  150  143  125  100  75  57  50  57  75  100  125  143 
 According to the data in the table, is $F$ a function of $R$? Is $R$ a function of $F?$ Explain.
 Are either $R$ or $F\,$ functions of $t$? Explain.
Comments
Log in to commentLloyd Allen says:
about 4 yearsIn order for this fertile task to address Process #3, I'd really like to explicitly request an explanation from the student. Otherwise, this is a task that could be scored by machine with answer "no, no, yes". I believe that good mathematicians tend to answer exactly the question that is asked, and I'd want my students' answers to look more like what has been submitted as the solution.
Kristin says:
about 4 yearsDone!
Lloyd Allen says:
about 4 yearsThanks!