The restaurant

Task

A restaurant is open from 2 pm to 2 am on a certain day, and a maximum of 200 clients can fit inside. If $f(t)$ is the number of clients in the restaurant $t$ hours after 2 pm that day,

1. What is a reasonable domain for $f$?
2. What is a reasonable range for $f$?

IM Commentary

The purpose of this task is to get students thinking about the domain and range of a function representing a particular context. Often when a function is being used to model a context, the expression for the function has a larger domain and range than is reasonable for the context. Asking students to focus on a function for which there is no formula focuses attention on the context itself.  Note that in many contexts, there are multiple plausible sets that one could choose for the domain (and/or the range), and the solution to the current task provides such an example.  

This task is adapted from Algebra: Form and Function, McCallum et al., Wiley 2010.

Solution

1. The input to $f$ is the number of hours after 2 pm. The restaurant is open from 2 pm to 2 am, so a reasonable domain for $f$ is $0 \le t \le 12$. Alternatively, it would be reasonable to assert that the domain is $0 \le t < 24$ with the understanding that $f(t)=0$ for $12<t<24$. 
2. The output of $f$ is the number of clients in the restaurant. The restaurant holds a maximum of 200 clients, so a reasonable range for $f$ is any whole number from 0 to 200.