Contextualization

Mathematics is a living, dynamic discipline and an intrinsic part of our daily lives, and one of the fundamental topics that plays a crucial role in various applications is the concept of 'Function and Domain'.

A function is a mathematical concept that consists of a rule that relates each element of one set to a single element of another set. The Domain is the set of all values that can be input into the function. In other words, it is the set of all possible input values that the function can accept.

The study of functions and domains allows for an understanding of how one variable depends on another and is the basis for many fields of knowledge, including physics, engineering, economics, computer science, among others.

Introduction

The domain of a function is the set of all possible inputs for that function. In simpler terms, it is the set of all possible values that the independent variable (x) can have in the function. To find the domain of a function, we need to consider the types of numbers that the function can accept without issues.

Functions with square roots, for example, cannot have negative numbers under the radical. Therefore, the domain of this function would be all real numbers greater than or equal to zero. Similarly, functions with variables in the denominator cannot have values that make the denominator equal to zero, as this would lead to division by zero, which is undefined in mathematics.

Understanding the concept of functions and domains is crucial for advancing in many fields. For example, in engineering, it is necessary to understand how one variable can affect another.

In economics, functions can be used to model and predict consumption and production behaviors. In computer science, functions are used in everything from creating algorithms to data representation.

The following links are suggested for a deepening and/or review of the concepts:

• Function Concept
• What is the Domain of a Function?
• Injective, Surjective, and Bijective Functions
• Real Functions of Real Variable

Practical Activity

Activity Title: 'Heading to Infinity: Exploring Functions and Domains'

Project Objective:

The 'Heading to Infinity' project aims to stimulate the understanding and appreciation of mathematical concepts of functions and their domains. Additionally, the project aims to promote student collaboration in solving complex problems, develop written and oral communication skills, and promote efficient time management.

Detailed Project Description:

Each group of 3 to 5 students will be challenged to:

1. Choose 5 different functions (real functions of real variable, quadratic functions, exponential functions, logarithmic functions, and trigonometric functions).

2. For each chosen function, students must determine its domain.

3. Apply these functions and domains in real-world scenarios. For example, choose a representative function that models an economic scenario, a function that models a physical phenomenon, etc.

4. Explain, in a clear and concise manner, the chosen function, its domain, the application, and the reason for this choice.

5. Create an attractive visual presentation of the results, making it clear how the chosen functions and domains apply to real-world situations.

Required Materials:

Students will need:

• Internet access for research
• Writing material
• Presentation software (e.g., PowerPoint, Google Slides, Prezi)
• Software tool for graphical representation of functions (e.g., Geogebra, Desmos)

Detailed Step-by-Step for Activity Execution:

1. Form groups of 3 to 5 students.

2. Each group should choose 5 functions of different types.

3. Find the domain for each chosen function.

4. Research possible applications of these functions in the real world.

5. Prepare a written document (project) according to the given guidelines.

6. Develop a clear and concise visual presentation of the results obtained and applications in the real world.

Project Submission:

Students must submit:

1. A written document in the form of a report containing:

   • Introduction to the theme (emphasis on the importance of functions and the identification of domains)

   • Development, which should contain a detailed description of the chosen functions, their domains, and chosen applications.

   • Conclusion, which should distill the project's learnings, the connections made in various disciplines (e.g., mathematics and economics, physics, etc.), and the benefits of teamwork collaboration.

   • Bibliography with all sources consulted in the research and project development process.

2. An attractive visual presentation showing the functions, domains, and their practical applications in a clear and concise manner.

Students should keep in mind that the written report and the visual presentation are complementary parts of the same project. The report provides a detailed and complete view of the project, while the visual presentation provides a concise and attractive overview of the project's findings and achievements.

By the end of this project, students will have not only deepened their technical knowledge of functions and domains but also developed valuable research, collaboration, communication, and time management skills.