Q: Formal Definitions and Problem Domain

The problem presents a specific case in the field of Functional Equations where we must determine all functions ##f: \mathbb{R} \to \mathbb{R}## satisfying the identity ##f(x+y) = f(x) + f(y) + 2xy## for all real numbers ##x## and ##y##. This equation is non-homogeneous because of the additive term ##2xy##, which distinguishes it from the basic Cauchy additive model. The domain and codomain are the set of real numbers, which implies that the function must be well-defined for all possible continuous values across the entire number line. Such problems require a rigorous examination of the structural mapping properties to ensure that the final result remains consistent regardless of the specific input values selected during the analytical process.

Q: Uniqueness and the Absence of Alternative Forms

A critical question in the study of Functional Equations is whether the identified solution is the only possible one. The uniqueness of ##f(x) = x^2 + x## follows directly from the uniqueness of the solution to Cauchy’s equation ##g(x+y) = g(x) + g(y)##. If ##g(x)## is required to be continuous or if we restrict ourselves to polynomials, then ##g(x) = cx## is the only possible form. Since the original problem is stated for all real numbers without additional pathological constraints, the standard analytical approach dictates that the linear solution is the only one that satisfies the requirements of a well-behaved real-valued function.

Q: Historical Context and Advanced Generalizations

The study of Functional Equations dates back to the early 19th century, with Augustin-Louis Cauchy being the first to rigorously analyze the additive identity that now bears his name. The specific equation ##f(x+y) = f(x) + f(y) + 2xy## can be seen as a variation of the Pexider equation or a specific case of the D’Alembert functional identity. Such forms frequently appear in the characterization of polynomial functions and are used in physics to describe properties like displacement under constant acceleration. The quadratic term effectively represents the interaction between variables that characterizes second-order systems in various scientific disciplines.

Question 1

Formal Definitions and Problem Domain

Accepted Answer

The problem presents a specific case in the field of Functional Equations where we must determine all functions ##f: \mathbb{R} o \mathbb{R}## satisfying the identity ##f(x+y) = f(x) + f(y) + 2xy## for all real numbers ##x## and ##y##. This equation is non-homogeneous because of the additive term ##2xy##, which distinguishes it from the basic Cauchy additive model. The domain and codomain are the set of real numbers, which implies that the function must be well-defined for all possible continuous values across the entire number line. Such problems require a rigorous examination of the structural mapping properties to ensure that the final result remains consistent regardless of the specific input values selected during the analytical process.

Question 2

Structural Inspection and Initial Point Evaluation

Accepted Answer

To begin the systematic evaluation of the function, we first investigate the behavior of ##f## at the origin. By substituting ##x = 0## and ##y = 0## into the original functional identity, we obtain the following relationship:
###f(0+0) = f(0) + f(0) + 2(0)(0)###
This simplifies to the equality ##f(0) = 2f(0)##, which uniquely implies that the function must pass through the origin such that ##f(0) = 0##. This initial finding is crucial as it establishes the intercept for the function and simplifies subsequent substitutions, particularly when attempting to determine parity or when performing shifts along the horizontal axis to test for linearity.

Question 3

Decoupling via Auxiliary Function Transformation

Accepted Answer

The core challenge in solving this problem lies in the presence of the bilinear term ##2xy##, which prevents a direct application of standard additive proofs. To resolve this, we hypothesize that the function ##f(x)## contains a quadratic component ##x^2##. This is motivated by the fact that the expansion of the square of a sum, ##(x+y)^2 = x^2 + y^2 + 2xy##, perfectly mirrors the non-homogeneous part of our equation. We therefore define an auxiliary function ##g(x)## designed to subtract this quadratic behavior from ##f(x)##. Let us define:
###g(x) = f(x) &#8211; x^2###
This transformation is intended to simplify the functional identity by shifting the complexity into a new variable space.

Question 4

Solving the Linear Component and Point Consistency

Accepted Answer

With the relationship ##g(x+y) = g(x) + g(y)## established, our next objective is to determine the specific form of ##g(x)## using the boundary condition provided in the problem statement. We know that ##f(1) = 2##. We can relate this back to our auxiliary function ##g(x)## using our initial definition ##g(x) = f(x) &#8211; x^2##. By substituting ##x = 1## into this expression, we calculate the value of ##g(1)## as follows:
###g(1) = f(1) &#8211; (1)^2 = 2 &#8211; 1 = 1###
This gives us a specific data point for the additive function ##g##, namely that the image of unity under this mapping is exactly one.

Question 5

Validation via Substitution and Algebraic Expansion

Accepted Answer

To verify the candidate solution ##f(x) = x^2 + x##, we must check if it satisfies the identity ##f(x+y) = f(x) + f(y) + 2xy## for all ##x, y \in \mathbb{R}##. We begin by evaluating the left-hand side of the equation for our specific function:
###f(x+y) = (x+y)^2 + (x+y) = x^2 + 2xy + y^2 + x + y###
This expansion is straightforward and provides the baseline for comparison. Next, we evaluate the right-hand side of the original equation by summing the images of ##x## and ##y## and adding the bilinear term:
###f(x) + f(y) + 2xy = (x^2 + x) + (y^2 + y) + 2xy###
By rearranging the terms on the right-hand side, we obtain ##x^2 + 2xy + y^2 + x + y##.

Question 6

Uniqueness and the Absence of Alternative Forms

Accepted Answer

A critical question in the study of Functional Equations is whether the identified solution is the only possible one. The uniqueness of ##f(x) = x^2 + x## follows directly from the uniqueness of the solution to Cauchy’s equation ##g(x+y) = g(x) + g(y)##. If ##g(x)## is required to be continuous or if we restrict ourselves to polynomials, then ##g(x) = cx## is the only possible form. Since the original problem is stated for all real numbers without additional pathological constraints, the standard analytical approach dictates that the linear solution is the only one that satisfies the requirements of a well-behaved real-valued function.

Question 7

Cauchy’s Equation and the Necessity of Regularity

Accepted Answer

The transformation of the original problem into the Cauchy Functional Equation ##g(x+y) = g(x) + g(y)## brings us into the realm of classical analysis. While the equation looks simple, its behavior over the real numbers is a subject of significant mathematical depth. For rational numbers ##q \in \mathbb{Q}##, it is a proven theorem that any additive function must satisfy ##g(q) = cq## for some constant ##c##. This is established through induction for natural numbers, extension to integers, and finally to fractions. However, moving from the rational domain to the full real domain ##\mathbb{R}## requires an additional assumption, often referred to as a regularity condition, to ensure the function remains linear.

Question 8

Historical Context and Advanced Generalizations

Accepted Answer

The study of Functional Equations dates back to the early 19th century, with Augustin-Louis Cauchy being the first to rigorously analyze the additive identity that now bears his name. The specific equation ##f(x+y) = f(x) + f(y) + 2xy## can be seen as a variation of the Pexider equation or a specific case of the D’Alembert functional identity. Such forms frequently appear in the characterization of polynomial functions and are used in physics to describe properties like displacement under constant acceleration. The quadratic term effectively represents the interaction between variables that characterizes second-order systems in various scientific disciplines.

JUPITER SCIENCE

Functional Equations: Solving f(x+y) = f(x) + f(y) + 2xy

Calculating Root Power Sums: Viete’s Formulas and Symmetric Sums

Infinite Geometric Series: Detailed Solution and Convergence Analysis

Complex Number Evaluation: Solving ##z^3 + 8 \text{ for } z = 1 + i\sqrt{3}##

Simplify ## \frac{x^2 – 9}{x^2 + 5x + 6} ##

Linear Equation with One Variable: Technical Solution and Analysis

What is a Plane? A Simple Explanation

Understanding Lines: What is a Line in Geometry?

Understanding the Concept: What is a Point?

Euclidean Geometry Explained: A Beginner’s Guide

Pythagorean Theorem Explained: History Proof Applications & Problems

Conic Sections: The Geometry of Curved Beauty

Understanding Ordered Pairs: Definition Examples and Applications

Solving Polynomial Equations: Finding q(uv) = 0

Polynomial Matrix Decomposition: A Step-by-Step Guide

Understanding Inequalities in Harmonic Homogeneous Polynomials

Categories

Recent Posts

Site Info

Social