Experience First

In this lesson, students consider various costs and distribution of costs to reason about arithmetic combinations of functions. Although the idea of adding (or subtracting or multiplying or dividing) functions is not novel for students, interpreting such situations in context can be more challenging. For example, in question 4, students must interpret what is meant by dividing the total cost function by the number of paying attendees. In question 2, if it does not come up naturally in student conversation, push students to consider that even though there doesn’t seem to be a cap on attendees established by the caterer, the rental space is limited, so this is the restriction that we must oblige with.

Furthermore, in question 3b and 5 students are asked to reason about how the given scenarios affect the reasonable domain of a function. While it is possible (but not probable) that the number of attendees is 12 (just the student council) and thus that the number of paying attendees is 0, this breaks down when we find the cost per paying attendee function, since we can’t divide by 0--who would pay for the prom? This is meant to highlight that dividing two functions often creates additional restrictions on the domain.

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Formalize Later

While the idea of combining functions using the four operations is pretty intuitive for students, students sometimes struggle to see how the domain is affected in a combination, particularly in a quotient. The idea is that even though functions f and g may have certain restrictions on their domain, when dividing the two functions, particular attention must be given to confirm that the denominator function does not output a value of 0. 

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For sums, differences, and products, the idea of “the most restrictive domain” can be explained with an analogy. I tell students that if they are having two friends over for dinner and one is gluten free and the other is lactose intolerant, then they must work to cook food that contains neither gluten nor lactose, a requirement that is more strict than either of the individual food allergies.

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In AP Calculus, it is often emphasized that all equivalent expressions are accepted. It may be advantageous to adopt this view in Precalculus as well, accepting expressions of combinations of functions where the like terms are not added, or the binomials are not multiplied out, for example.