Context

In mathematics, a fundamental concept is that of a function. A function is a special relation that associates a value from one set to another value in another set. For a function to be well-defined, each value in the starting set (called the domain) must be associated with exactly one value in the target set (called the codomain). The image of a function is the subset of the codomain that is actually reached by the function.

Understanding these concepts is crucial not only in mathematics but in various disciplines such as physics, economics, engineering, among others. They are used to model and solve real-world problems. For example, in economics, a demand function can relate the price of a good (domain) to the quantity demanded of that good (image), the universe of possible quantities demanded is the codomain.

In more advanced studies, a good understanding of these concepts is essential to study topics such as limits, derivatives, and integrals. For 1st-year high school students, understanding the concept of function, codomain, and image is an important step in developing more complex mathematical skills in the future.

To delve deeper into the topic, we recommend the following sources:

1. Book: Gonçalves, Manoel M. & Custódio, José E. (2017). Matemática e raciocínio: uma introdução à Matemática Superior.
2. Website: Só Matemática: On this website, you will find a detailed explanation, with interactive examples, about functions, domain, image, and codomain.
3. Video: Matemática Rio Channel with Prof Rafael Procopio: This is a great audiovisual resource for those who prefer a video format for learning. Prof. Rafael provides an excellent introduction to functions, domain, codomain, and image.

Practical Activity

Activity Title: "Domain in the Functions Park"

Project Objective:

The purpose of this project is to gain a practical understanding of functions, especially the concept of codomain and image of a function. Students should learn how to calculate the image of a function and verify when the codomain is equal to the image.

Detailed Project Description:

Students will create a "Functions Park" where each attraction represents a different function. These attractions (functions) will receive inputs (domain values) and produce outputs (image values). The set of all possible outputs in all attractions constitutes the codomain of the park.

Required Materials:

• Graph paper
• Ruler
• Pencil or pen
• Calculator

Detailed Step-by-Step Guide for the Activity:

1. Group Formation and Planning: Form groups with 3 to 5 students each. Each group should plan the "Functions Park" and choose 3 different functions to be the attractions of the park.

2. Definition of Functions: Define the domain, image, and codomain of the three chosen functions. Draw each function on the graph paper.

3. Creation of the "Functions Park": For each function, decide the domain value (input) and calculate the corresponding image value (output). Remember, the codomain is the set of all possible outputs, not just those you calculated.

4. Report Elaboration: Each group should prepare a report where they explain:

   • Introduction: the relevance of learning about functions, codomain, and image and the purpose of the project.
   • Development: the theory of functions, codomain, and image, the process of creating the "Functions Park", the methodology used, and the results obtained.
   • Conclusions: the learnings, the difficulties encountered, and how they were overcome.
   • Bibliography used: all sources consulted for the project.

Project Submission:

The final work will consist of the written report (Introduction, Development, Conclusions, and Bibliography) and the "Functions Park". Students must describe in detail in the report how the park represents the functions, domain, image, and codomain and how the calculations were made. The park will be evaluated for its visual representation and mathematical accuracy.

Students should explain in the report how the activities carried out helped to understand the concepts of functions, codomain, and image, as well as detail how the teamwork process was, which social skills were developed (communication, problem-solving, time management, etc.), and how they were used.