Unveiling Functions: Theory and Practice for the Real World

Objectives

1. Understand the concept of function, identifying that each input element has only one output.

2. Explore the dependency relationships between two variables through practical examples.

3. Learn to mathematically represent functions, such as in the form y=2x+3.

Contextualization

Mathematical functions are present in various situations of our daily lives. From calculating the average speed of a car to predicting plant growth, functions help us understand and predict behaviors. For example, when using a weather forecasting app, we are utilizing mathematical functions to know the temperature for the following day. Understanding how one variable can affect another is essential for efficiently solving practical problems.

Relevance of the Theme

The topic of functions is extremely relevant in the current context, as its application extends to areas such as computer programming, data analysis, engineering, and finance. Understanding functions is essential for developing problem-solving skills and critical thinking, which are highly valued in the job market.

Concept of Function

A function is a dependency relationship between two variables, where for each input value (x) there is a unique output value (y). It is a mathematical tool that allows modeling and predicting behaviors in various everyday situations.

• Each input element has only one output.

• Allows modeling relationships between variables.

• Essential for solving practical problems.

Dependency Relationships Between Variables

The dependency relationship between variables is the basis of the function's definition. Here, one variable depends on the other, meaning that the value of y depends on the value of x. This is crucial for understanding how changes in one variable affect the other.

• The value of y depends on the value of x.

• Allows predicting outcomes and behaviors.

• Fundamental for modeling real situations.

Graphical Representation of Functions

The graphical representation of functions is a visual way to show how y values vary according to x values. Graphs facilitate understanding of the relationships between variables and allow for identifying patterns and trends.

• Facilitates visualization of relationships between variables.

• Allows identifying patterns and trends.

• Essential for data analysis and problem-solving.

Linear Functions and Their Applications

Linear functions are those whose graphical representation is a straight line. They are used to model situations where there is a direct and constant relationship between two variables, such as the constant speed of a car or the cost of a service that varies linearly with time.

• Graphical representation is a straight line.

• Models direct and constant relationships between variables.

• Applicable in various practical situations, such as constant speed and linear costs.

Practical Applications

• Weather Forecasting: Mathematical models that use functions to predict weather conditions based on historical and current data.
• Computer Programming: Use of functions to create algorithms and solve complex problems efficiently.
• Engineering: Application of functions to model and simulate the behavior of structures and systems under different conditions.

Key Terms

• Function: Dependency relationship where each input has a unique output.

• Variable: A value that can change and that is used to represent data in a function.

• Graphical Representation: Visual way to show how y values vary with x values.

• Linear Function: Function whose graphical representation is a straight line, showing a linear relationship between two variables.

Questions

• How can mathematical functions be applied in your daily life?

• In what way can understanding dependency relationships between variables help in solving practical problems?

• What other areas of knowledge, besides mathematics, utilize the concept of functions? Explain how.

Conclusion

To Reflect

Functions are powerful tools that allow us to understand and model dependency relationships between variables. They are present in various areas of our daily lives, from weather forecasting to computer programming. Understanding these relationships is essential for solving practical problems efficiently. During our activities, we had the opportunity to visualize and construct graphical representations of functions, which helped us better understand how one variable can affect another. By applying these concepts in real situations, we developed critical skills that are highly valued in the job market.

Mini Challenge - Creating Everyday Functions

In this challenge, you will create a function that models a practical situation from your daily life.

• Think of a daily situation where one variable depends on another. For example, the cost of a taxi that has a base fare and a cost per kilometer traveled.
• Write the function that represents this situation. For example, Cost = 5 + 2 * (number of kilometers).
• Create a table of values for different inputs. For example, for 1 km, 2 km, 3 km, etc.
• Draw the graph of the function, showing how the cost varies according to the number of kilometers.
• Explain how this function can be useful to predict or plan something in your daily life.