Contextualization

Before we move on to the practical part, let's take a closer look at what a function is. In mathematics, a function is a rule that relates each element of one set to exactly one element of another set. It can be described by a mathematical formula and is often represented by f(x), where 'x' is the input set and 'f(x)' is the output set. But remember, a function must have one and only one output for each input!

Imagine you are cooking a dish. The ingredients you use are like the inputs of a function, and the finished dish is like the output of the function. If you change a single ingredient, the result, that is, the taste of the dish, will probably change. Similarly, in a function, if you change the input, the output will be different. Thus, we can say that a function is a kind of 'processing machine' that transforms inputs into outputs.

Connection to the Real World

You may be wondering, 'where do we use this thing called a function in real life?'. Well, functions are present in virtually all areas of our lives! For example, when you are driving and using the car's GPS, you are using a function that takes your current location as input and the next direction to be followed as output.

If you play video games, you may have noticed that games become more challenging as you progress. This is done with the help of a function that takes into account the player's current level to determine the difficulty of the next level. Therefore, functions are a fundamental part of exact sciences and technology, and not knowing them would be like trying to read a book without knowing the letters of the alphabet!

Recommended Resources

To deepen your knowledge of the function concept, we recommend the following study sources. Remember, it is important to understand the theory well before moving on to the practical part!

1. Book 'Fundamentos de Matemática Elementar - Vol. 1: Conjuntos, Funções - Gelson Iezzi, Osvaldo Dolce, Carlos Murakami'.
2. Khan Academy articles on Functions: What is a function?
3. Video lessons from the YouTube channel Matemática Rio with Prof Rafael Procópio on Functions: Functions - What is a Function

Practical Activity

Activity Title: 'The Kitchen of Functions'

Project Objective:

• Learn to define a function.
• Check the existence conditions of a function: only one output for each input and all components of the domain have an output.

Detailed Project Description

Your mission, as a group of 3 to 5 students, will be to create a 'functions menu'. Each dish on this menu is a function, the ingredients are the inputs, and the result of the dish is the output of the function. For each dish (function), there will be only one way to prepare it (one output), and each ingredient (input) will be used only once.

To represent the dishes (functions), you will need to create graphs that show the relationships between the ingredients (inputs) and the resulting dish (output). It is important to remember that each dish (function) must have one and only one output for each input.

The project aims not only at understanding the function concept but also at developing skills such as teamwork, communication, time management, and creative thinking.

Required Materials

• Paper and colored pens for creating the menu and graphs.
• Internet access for research.

Detailed Step-by-Step Guide for the Activity

Step 1:

Choose between 3 to 5 different dishes to create the functions. Remember, each dish is a function!

Step 2:

Make a list of ingredients for each dish (function). These ingredients are the function's inputs.

Step 3:

Write down how to prepare each dish (function). This will be the result or output of the function.

Step 4:

Create a graph for each dish (function) showing the relationship between the ingredients (inputs) and the resulting dish (output).

Step 5:

Compile all your functions into a colorful and creative 'functions menu'. Each dish (function) should have a dedicated page with the list of ingredients (inputs), the preparation method (output), and the corresponding graph.

After completing the practical part, the next step will be to prepare a written report, following the detailed structure below:

1. Introduction: Contextualize the function concept based on your previous knowledge and theoretical study. Also, explain the objective of the 'The Kitchen of Functions' project.

2. Development: Explain the theory of functions and how they were applied in creating the 'functions menu'. Include details about the methodology used in the project, the process of creating the dishes (functions), and the discussion of the results obtained.

3. Conclusion: Review the main points of the project, highlight the learnings obtained, and conclude on the importance of functions in mathematics and everyday life.

4. Bibliography: List the resources used during the project, such as books, internet sites, videos, among others.

Remember: The work must be completed in one week, and it is expected that each student invest between two to four hours in the project.