Project: Functions in Action: Exploring Co-domain and Range

Contextualization

Functions are an extremely useful and versatile mathematical tool, applied in various fields of knowledge such as physics, economics, biology, computing, and many others. A function is a relation that associates each element of a set A (the domain) with exactly one element of a set B (the codomain). However, it is important to note that not all elements of the codomain are necessarily related to any element of the domain. Those that are form the function's range. In more practical terms, the domain of a function is the set of all possible input values (x), the codomain is the set where the output (f(x)) can potentially reside, and the range is the set of all actual values that the function takes on. Understanding the concepts of range and codomain of a function is essential to effectively use and interpret functions both in mathematics and in its applications. It allows us to understand the limits and possibilities of the function, reducing interpretation errors and opening the way for more complex and detailed analysis.

Introduction

In this project, we will explore the concepts of function, codomain, and range in more depth. We will learn to distinguish between codomain and range, and how to calculate the range of a function. Furthermore, we will explore how functions play a central role in various disciplines beyond mathematics, such as physics and economics. In addition to the theoretical study, the project also includes an exciting practical activity that will stimulate participants' creativity and collaborative spirit. Furthermore, in the elaboration of the final report, students will have the opportunity to improve their research, scientific writing, and time management skills. The project is based on the following resources that provide a good theoretical foundation:

• Book "Fundamentals of Elementary Mathematics Volume 1: Sets, Functions" - Gelson Iezzi.
• Video lessons from the channel "Matemática Rio com Prof Rafael Procopio" on YouTube (Note: the video lessons are in Portuguese and address the topic of functions in a simple and straightforward way).
• Website "Brasil Escola" in the section dedicated to mathematics, which contains a variety of articles on the topic and solved exercises.
• Platform "Khan Academy" which presents a series of lessons and interactive exercises on functions.

The combination of these resources will provide students with a solid understanding of the concepts and theories necessary to complete the project.

Practical Activity

Activity Title: "Functions in Action: Exploring Co-domain and Range"

Project Objectives:

1. Understand the concepts of function, codomain, and range.
2. Know how to calculate the range of a given function.
3. Verify when the codomain is equal to the range.
4. Improve the collaboration and communication skills between team members.
5. Develop research and application skills of theoretical concepts to practical problems.
6. Stimulate analytical and critical thinking through the interpretation and analysis of results.

Detailed Project Description:

Students, divided into groups of 3 to 5, will explore the proposed theme by developing and solving practical problems related to the theme "Function - Codomain and Range". The groups must create a portfolio that includes the theoretical part (foundation, problem example, and solution) and practice (problem created and solved by the group, with detailed explanation of the steps and conclusions).

In addition, to make the project interdisciplinary, students must choose a real application of the concept of functions in another discipline (e.g., physics, economics, chemistry) and explain how the use of functions, specifically the understanding of co-domain and range, is necessary to solve practical problems in that area. The project will last two weeks, during which each student will dedicate at least twelve hours to the project.

Required Materials:

1. Math books and/or reliable online sources for study and theoretical foundation.
2. Paper and pen for notes and problem solving.
3. Computer with internet access for research and final report preparation.
4. Text editing software for final report preparation (e.g., Microsoft Word, Google Docs).

Step-by-Step Instructions for the Activity:

1. Individual and group study of the concept of function, codomain, and range using the recommended resources. Each student should take notes and everyone should contribute with their doubts and observations to build a collective understanding.

2. After understanding the concepts, students should meet and create a mathematical problem exemplifying the application of these concepts. The problem's solution should be detailed, showing all the steps and justifying them with the studied theory.

3. The group should then create an original problem involving the concept of functions, codomain, and range. The problem's solution should also be detailed.

4. To incorporate interdisciplinarity, students should choose a discipline and a practical problem where the concept of functions is applied. They should explain how the function contributes to the solution and how the understanding of codomain and range is crucial.

5. Students should record all the project's steps, their notes, problems, solutions, and discussions in a final portfolio.

6. Finally, students should write a report in scientific article format containing:

   • Introduction: Contextualization, relevance, and application of the theme; project objective.
   • Development: Explanation of the studied theories, detailed description of the activity with methodology used; presentation and discussion of the results.
   • Conclusion: Recapitulation of the main points and conclusions about the project.
   • Bibliography: Bibliographic references used.

7. The group should review the portfolio and the report to ensure that all aspects of the project have been addressed and that the work is well written and formatted.

Project Deliverables

1. Portfolio with the entire study process and problem solving, including individual and group notes, problems created and solved, interdisciplinary application, and discussions.

2. Report in scientific article format, describing the group's experience in the project, the studied theories, the method used, and the conclusions about the project.

The project deliverables are interconnected. The development of the portfolio supports the construction of the final report by providing a record of the learning process and project development. The final report, on the other hand, uses the information from the portfolio to explain in a structured and detailed way the studied concepts, the activities carried out, the results obtained, and the meaning of these results.

To produce the report, students should follow the instructions for preparing a scientific article. The introduction should present in a concise way the theme and the objective of the project. The development should describe the studied theory, the methodology used, the activities carried out, and the results obtained. The conclusion should summarize the main points and explain what was learned, the difficulties encountered, and how they were overcome. The bibliography should reference all sources of information used in the project.