Context

Theoretical Introduction

Functions are one of the foundations of mathematics, permeating the study of various areas, from calculus to number theory. Within this topic, a fundamental classification is injective and surjective functions.

The injective function, also called an injective function, is one in which distinct elements of the domain correspond to distinct elements of the codomain. In other words, if two different values are input into the function, they will produce different results.

On the other hand, the surjective function is one in which every element of the codomain is the image of at least one element of the domain. That is, for any value we choose in the set of possible results, there is at least one value that, if input into the function, will produce that result as output.

Context

Injective and surjective functions have practical applications in various areas, such as physics, economics, biology, computer science, and many others.

For example, in physics, injective functions are used in fluid mechanics to ensure that different particles cannot occupy the same place at the same time. In economics, they are used to model the relationship between different goods and services and their prices, ensuring that different prices lead to different quantities of demand.

Surjective functions, in turn, are used in areas such as computer science, where algorithms that map inputs to outputs need to be able to produce all possible outputs.

Practical Activity

Activity Title: Challenge of Functions: Injective and Surjective Functions

Project Objective

This project aims to stimulate the learning of the concepts of injective and surjective functions through a practical challenge that involves the application of these concepts in real situations. Additionally, it aims to promote the development of teamwork, research, use of technologies, and writing skills.

Detailed Project Description

Students will be divided into groups of 3 to 5 people, and each group will receive a set of mathematical functions, distributed according to difficulty. The students must identify which of these functions are injective, which are surjective, and which are both injective and surjective. They must also justify their answers.

The groups will apply the concepts learned to model real-world situations. Each group must choose a real-world problem that involves the application of injective and surjective functions. They must mathematically model this situation, developing a detailed study of how the functions apply to the chosen situation.

Required Materials

• Mathematics books on functions
• Internet access for research
• Whiteboard or paper for drawing graphs
• Calculator

Activity Steps

1. Study: Students must first study the concept of injective and surjective functions, using the resources available in the materials link and others they find.
2. Identification: Each group will receive a list of functions and must determine whether each function is injective, surjective, or both, justifying their answers.
3. Problem Selection: Next, each group must select a real-world problem to which injective or surjective functions can be applied. This situation can be from any area - economics, physics, computer science, etc.
4. Modeling: Students will mathematically model the chosen problem through injective and/or surjective functions.
5. Report Development: Finally, students must prepare a detailed report explaining the entire process, from the initial study to the application of the concept to the chosen problem.

Project Deliverables

At the end, students must submit a written report containing:

1. Introduction: Contextualizing the chosen problem, relevance of injective and surjective functions, project objective.
2. Development: Theoretical discussion on injective and surjective functions, detailed description of the activities carried out, methodology adopted to solve the challenge, and results obtained.
3. Conclusions: Reflection on the learnings, difficulties and achievements, understanding reached about the functions, and applicability in future situations.
4. Bibliography: References of the study and research materials used.

The report should clearly express the students' understanding of the studied functions and their application, as well as the teamwork process and skills developed throughout the project.