Question

A restaurant is open from \(2 \mathrm{pm}\) to 2 am each day, and a maximum of 200 clients can fit inside. If \(f(t)\) is the number of clients in the restaurant \(t\) hours after \(2 \mathrm{pm}\) each day, (a) What is reasonable domain for \(f ?\) (b) What is a reasonable range for \(f ?\)

Short answer

Answer: (a) The reasonable domain of \(f(t)\) is \([0, 12]\). (b) The reasonable range of \(f(t)\) is \([0, 200]\).

Step-by-step solution

Determine the opening hours in terms of t hours after 2 pm

The restaurant is open from 2 pm to 2 am. We can convert this to hours after 2 pm by calculating the duration of time the restaurant is open. From 2 pm to 12 am, there are 10 hours, and from 12 am to 2 am, there are 2 more hours. So, the restaurant is open for a total of 12 hours.

Determine the domain of f(t)

Since the function \(f(t)\) is the number of clients in the restaurant t hours after 2 pm, the domain is the range of hours the restaurant is open. We found that to be 12 hours, so the domain of f(t) is from 0 hours (at 2 pm) to 12 hours (at 2 am). In interval notation, this is written as \(0 \le t \le 12\) or \([0, 12]\).

Determine the range of f(t)

The range of \(f(t)\) will be the range of the number of clients the restaurant can hold. Since the maximum capacity is 200 clients, the range of f(t) will be from 0 clients (assuming it is possible that the restaurant might be empty) to 200 clients (the maximum possible clients). In interval notation, this is written as \(0 \le f(t) \le 200\) or \([0, 200]\). (a) The reasonable domain of \(f(t)\) is \([0, 12]\). (b) The reasonable range of \(f(t)\) is \([0, 200]\).

Key concepts

Piecewise Functions

Piecewise functions are mathematical expressions defined by different rules or formulas over different intervals. Imagine trying to model the number of clients in a restaurant at different times of the day. The number might be low during early hours and peak around dinner time, dropping later again. A single function formula might not capture this variation effectively.

When dealing with such situations, piecewise functions become very useful. Each "piece" of the function describes behavior during a particular time interval.

For example, one piece might show an increase in client numbers during dinner hours, while another piece shows a decrease post-dinner. Analyzing real-world data often involves breaking down a continuous timeline into manageable sections, making piecewise functions a great tool.

Interval Notation

Interval notation is a concise way of writing the domain or range of a function. It describes the set of numbers lying between two endpoints.

The notation uses brackets and parentheses:

• Square brackets \( [ \) or \( ] \) indicate that the endpoint is included—called a "closed" interval.
• Parentheses \( ( \) or \( ) \) mean the endpoint is not included—called an "open" interval.

Let's look at our restaurant example.
The domain, written as \([0, 12]\), tells us that the function \( f(t) \) is valid from 0 to 12 hours, including both endpoints. Similarly, the range \([0, 200]\) includes all possible client numbers from 0 to 200.

Real-world Applications

Understanding functions, domains, and ranges through real-world examples can clarify abstract mathematical concepts.

In our restaurant scenario, defining the domain of \( f(t) \) helps manage operational hours, while the range reflects capacity limits. These are not just theoretical considerations; they make real impacts on customer service and planning.

By grasping these concepts, businesses can optimize their processes and better anticipate demand patterns.
With piecewise functions, managers can analyze different behaviors at various times—such as peak or off-peak hours—allowing for data-driven decision-making.

This shows how mathematics is intertwined with daily activities, making academic pursuits valuable beyond the classroom.