Princípios sobre a teoria das Equações Funcionais via aplicações e uma proposta de intervenção no ensino básico - Freire, Anderson Santos (2021)
The purpose of this text is to disclose the premises of the study on the most classic Functional Equations and Inequations, considering their relevance for the development of mathematics, also having the vision of spreading a proposal for research material that contributes to the improvement of teaching in this field of mathematics, which is little explored in Brazilian literature. We present throughout the writing, a brief discussion about the history of some scholars who made use of Functional Equations, showing aspects of solutions for certain standard Functional Equations, namely, Additive Cauchy Equation, Jensen Equation and Linear Functional Equation. Furthermore, we present a detailed study on the classes of solutions that characterize the Exponential, Logarithmic, Functional Equations, Cauchy Multiplicatives and the D’Alembert Equation. We emphasize that we were able to generalize, throughout the work, some Functional Equations, such as the Cauchy’s Additive Functional Equations, with the goal of looking for more complex solutions that satisfy the so-called Pexider and Vince Equations. We also explain a study of Functional Equations involving two variables, such as the Euler Equation and the Additive Cauchy Equation in two variables. We will also discuss certain special cases of a family of Functional Equations of a variable, called the Conjugation Equation, among these, the Schr¨oder Equation , the Abel Equation, and the B¨ottcher Equation. We will also show results on Functional Equations with multiple radicals and Polynomial equations, both proposed by the famous Indian mathematician Srinivasa Ramanujan. Finally, we will illustrate some applications of Functional Equations in Basic Education problems, more strictly, in questions from the Mathematics Olympiads, contained in the most varied events of this category, both in scope national and international.