Relevance of the Theme

Function: Introduction is the foundation of differential and integral calculus, one of the main areas of mathematics. The function is a fundamental concept that encompasses many other mathematical concepts. Understanding functions allows for the modeling of natural and social phenomena, the interpretation of graphs and tables, the prediction of behaviors, among other practical applications. Without functions, the world of mathematics would be monotonous and uniform, without the richness and complexity we find in reality.

Contextualization

The introduction to functions is a vital component of the first-year high school mathematics curriculum. Functions are often introduced to students after the completion of the study of real numbers, exponentiation and radical expressions, and first and second-degree equations. These areas provide a prior understanding that prepares students to deepen their knowledge in this new stage of study. At this stage, students move from a more concrete mathematics to a more abstract mathematics, and the concept of function is an important step in this transition.

Through the analysis of the behavior of functions, students will be able to understand and solve a variety of mathematical and practical problems. The introduction of functions opens up a vast conceptual landscape, in which students begin to visualize the relationships that exist between different quantities and the effects that changes in one quantity have on the others. This is a crucial step in the development of mathematical thinking.

Functions also provide the building blocks for the study of more advanced topics, such as trigonometry, calculus, and linear algebra. Therefore, it is essential for students to master this concept in a solid and complete manner. Understanding functions at this level will form the basis for further studies in mathematics and for practical application in numerous areas.

Theoretical Development

Components

• Definition of Function: A function is a correspondence between two sets, usually called the domain and codomain (or image). To each element of the domain, a unique element of the codomain is associated. It can be represented in various ways, such as by a rule of formation, by a table, by a graph, or by a formula.

• Elements of the Function: The function is composed of four main elements: the domain (set of all values of x for which the function is defined), the image (set of all values of y that the function can take), the dependent variable (y, whose value is determined by the value of x), and the independent variable (x, whose value is freely chosen).

• Properties of Functions: Functions have specific properties. Some examples include: parity (a function is even if f(-x) = f(x) for all x in its domain, while it is odd if f(-x) = -f(x)), periodicity (a function is periodic if f(x + T) = f(x) for all x in its domain, where T is the period), and limits (which determine the behavior of the function near specific points of its domain).

• Classification of Functions: Functions can be classified according to their characteristics. Some common examples are: constant function (f(x) = k, where k is a constant), linear function (f(x) = a·x + b, where a and b are constants, with a ≠ 0), quadratic function (f(x) = a·x² + b·x + c, where a, b, and c are constants, with a ≠ 0), among others.

Key Terms

• Dependent and Independent Variables: In a function, the dependent variable (represented by y) is a function of the independent variable (represented by x). The value of y is determined by the value of x.

• Domain and Image: The domain is the set of all values that x can take, while the image is the set of all values that y can take.

• Constant, Linear, and Quadratic Functions: These are specific types of functions that have formulas characterized by their structure. Understanding these terms is crucial for the classification of functions.

Examples and Cases

• Example 1: Let f(x) = 2x + 1. In this function, the domain is the set of all real numbers, the image is the set of all real numbers, and the value of y is always twice the value of x, plus 1.

• Example 2: Let g(x) = x² + 3. In this function, the domain is the set of all real numbers, the image is the set of all real numbers greater than or equal to 3, and the value of y is the value of x squared, plus 3.

• Example 3: Let h(x) = 4. In this function, the domain is the set of all real numbers, the image is the set containing only the number 4, and the value of y is always 4, regardless of the value of x. Therefore, h(x) is a constant function and its graph is a straight line segment parallel to the x-axis, at a distance of 4.

Detailed Summary

Key Points

• Understanding the Function: The concept of function is the backbone of mathematics. It is a way to relate a set of inputs (the domain) to a corresponding set of outputs (the image). This allows for the modeling and solving of an incredibly wide range of problems.

• Elements of the Function: Each function consists of four main components: domain, codomain, dependent variable, and independent variable. Understanding these elements allows for a precise and complete description of any function.

• Classification of Functions: Functions can be classified in various ways, such as constant, linear, and quadratic functions. Each type of function has its own characteristics and behaviors, which are crucial for its analysis and use.

• Representation of Functions: Functions can be represented in various forms, such as a rule of formation, a table, a graph, or a formula. The ability to understand and convert between these different representations is fundamental.

Conclusions

• Importance of Functions: Functions are an essential tool for solving mathematical and practical problems. They allow us to understand how one quantity depends on another and how they vary together.

• Applicability of Functions: Functions have applications in almost all areas of mathematics and their applications extend beyond mathematics, being used in statistics, physics, economics, computer science, engineering, and many other disciplines.

• Next Steps: A solid understanding of functions sets the stage for exploring more advanced mathematical concepts, such as trigonometry, calculus, and linear algebra.

Suggested Exercises

1. Identify the domain, image, and types of functions of the following expressions:

   • f(x) = x + 3
   • g(x) = x² - 2
   • h(x) = 5

2. Convert the following expressions into a table and a function graph:

   • f(x) = 2x - 1
   • g(x) = x² + 2x
   • h(x) = 4

3. Sketch the graph of the following functions and identify their main characteristics (intercepts, concavity, domain, image, etc.):

   • f(x) = 2x + 1
   • g(x) = x² - 2
   • h(x) = 4