A functional equation is an equation in which the unknowns are functions rather than numbers or variables. The aim is to determine all functions, typically with specific domains and codomains, that satisfy the given equation for all permissible values of the independent variable(s). Functional equations arise naturally throughout algebra, combinatorics, and mathematical analysis, especially in the context of olympiad and competitive examinations.

Formal Definition and Classification of Functional Equations

Given two non-empty sets $A$ and $B$, a functional equation is an equality involving a function $f: A \rightarrow B$, together with its values at various arguments (possibly expressed through algebraic, trigonometric, or other transformations), that must hold for all $x$ (and possibly other variables) in $A$. The central problem is the explicit determination of all such functions $f$ which satisfy the prescribed relation.

Functional equations are classified based on the nature of the function involved and the form of the equation. Standard classifications include Cauchy's equation, Jensen's equation, quadratic equations, additive/multiplicative equations, and symmetry-based functional equations. These can be further subdivided based on whether the relation is linear, quadratic, periodic, or exhibits other algebraic or analytic structures. For foundational discussions, refer to Sets, Relations, and Functions.

Representation of Fundamental Functional Equations over $\mathbb{R}$

The simplest functional equations involve linear transformations of the argument. For example, for $f:\mathbb{R}\to\mathbb{R}$, the equation $f(x + c) = g(x)$ for a fixed $c$ and a known function $g$ is a standard prototype. Another common representation is $f(ax + b) = h(x)$ for constants $a$, $b$, and a given $h(x)$, which leads to consideration of solutions via substitution and invariance.

Functional equations often appear in terms expressing symmetry. For example, the equation $f(x) + f(-x) = 2$ expresses an even-odd decomposition about the origin, while $f(x+y) = f(x)f(y)$ relates to exponential functions by characterizing multiplicative properties.

Algebraic Techniques in Solving Functional Equations

A primary method for solving functional equations is suitable substitution of variables to reduce the equation to a simpler form, frequently producing auxiliary equations. For example, for the equation $f(x+1) = 2f(x) + 1$, substituting $x \to x-1$ yields a new equation for $f(x)$ that can be solved recursively.

Symmetry analysis is also fundamental. By examining configurations such as $x \leftrightarrow y$ or $x \to -x$, one can isolate specific properties, e.g., parity (evenness/oddness), periodicity, or invariance, facilitating reduction of the problem to a known case.

In advanced cases, injectivity (one-to-one), surjectivity (onto), or bijectivity of the unknown function needs to be established or leveraged. Such properties are central in distinguishing admissible solutions, particularly for equations defined on open intervals or restricted domains. The distinction between injective, surjective, and bijective mappings is explored in depth on the Functions and Its Types page.

Structure and Graphs of Canonical Functional Equations

Certain standard forms of functions repeatedly arise as solutions to functional equations:

The linear function $f(x) = mx + c$ provides the general solution to additive equations of the form $f(x+1) = f(x) + d.$

The quadratic function $f(x) = ax^2 + bx + c$ solves recurrence-type equations involving squares of variables, such as $f(x+1) - 2f(x) + f(x-1) = k.$

Highly structured equations may admit solutions in terms of absolute value ($f(x) = |x|$), reciprocal ($f(x) = 1/x$), logarithmic ($f(x) = \ln x$), exponential ($f(x) = a^{x}$), or trigonometric ($f(x) = \sin x$, $f(x) = \cos x$) functions, depending on the algebraic properties displayed within the equation.

Complete Solution of a Simple Shift-Type Functional Equation

Given: $f(x+3) = x^2 + 8x + 16$ for all $x\in\mathbb{R}$

Step 1: Substitution — Let $y = x-3$. Then $x = y+3$. Substitute into the original equation to express $f(x)$:

$f((x-3) + 3) = (x-3)^2 + 8(x-3) + 16$

Step 2: Simplification — The left side becomes $f(x)$ and the right side expands as follows:

First, expand $(x-3)^2$:

$(x-3)^2 = x^2 - 6x + 9$

Secondly, expand $8(x-3)$:

$8(x-3) = 8x - 24$

Sum the terms together:

$f(x) = (x^2 - 6x + 9) + (8x - 24) + 16$

Combine like terms:

$f(x) = x^2 - 6x + 9 + 8x - 24 + 16$

$f(x) = x^2 + (-6x + 8x) + (9 - 24 + 16)$

$f(x) = x^2 + 2x + 1$

Result: $f(x) = (x+1)^2$

Determining Functions Satisfying Inversive Relations

Given: $f:\mathbb{R}\setminus\{0\} \to \mathbb{R}$ and $f(x) + 3 f\left(\dfrac{1}{x}\right) = x^2$ for all $x \in \mathbb{R}\setminus\{0\}$

Step 1: Substitution — Replace $x$ by $\dfrac{1}{x}$ everywhere to obtain a second equation:

$f\left(\dfrac{1}{x}\right) + 3f(x) = \left(\dfrac{1}{x}\right)^2 = \dfrac{1}{x^2}$

Step 2: System of Equations — Now two linear equations in $f(x)$ and $f\left(\dfrac{1}{x}\right)$:

$\begin{align*} \quad &(1)\quad f(x) + 3f\left(\dfrac{1}{x}\right) = x^2 \\ \quad &(2)\quad f\left(\dfrac{1}{x}\right) + 3f(x) = \dfrac{1}{x^2} \end{align*}$

Step 3: Solve for $f(x)$

Multiply equation (1) by $3$:

$3f(x) + 9f\left(\dfrac{1}{x}\right) = 3x^2$

Subtract equation (2) from this new equation:

$(3f(x) + 9f\left(\dfrac{1}{x}\right)) - (f\left(\dfrac{1}{x}\right) + 3f(x)) = 3x^2 - \dfrac{1}{x^2}$

$3f(x) + 9f\left(\dfrac{1}{x}\right) - f\left(\dfrac{1}{x}\right) - 3f(x) = 3x^2 - \dfrac{1}{x^2}$

$8f\left(\dfrac{1}{x}\right) = 3x^2 - \dfrac{1}{x^2}$

$f\left(\dfrac{1}{x}\right) = \dfrac{3x^2 - \dfrac{1}{x^2}}{8}$

$f\left(\dfrac{1}{x}\right) = \dfrac{3x^2}{8} - \dfrac{1}{8x^2}$

Now, replace $x$ by $\dfrac{1}{x}$:

$f(x) = \dfrac{3\dfrac{1}{x^2}}{8} - \dfrac{1}{8}x^2 = \dfrac{3}{8x^2} - \dfrac{x^2}{8}$

Result: $f(x) = \dfrac{3}{8x^2} - \dfrac{x^2}{8}$, valid for all $x \in \mathbb{R}\setminus\{0\}$

Method of Recurrence in Functional Equations

A functional equation may encode recurrence, such as $f(x+1) = f(x) + k$, where $k$ is a constant. This structure allows the transformation of the problem into solving a recurrence relation analogous to those studied in discrete mathematics and combinatorics. The complete explicit form must be derived using initial conditions and iteration, analogous to the processes detailed in Differential Equations.

Canonical Examples Illustrating Classification by Structure

Example: Additive Cauchy Functional Equation

Given $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$, and if $f$ is assumed continuous or otherwise regularized, then $f(x) = kx$ for some constant $k$. The critical point is that without regularity hypotheses, all additive solutions cannot be classified explicitly; the assumption of continuity or monotonicity ensures linearity.

Example: Multiplicative Equation

Given $f(xy) = f(x)f(y)$, the solution over positive reals, together with suitable regularity, is $f(x) = x^k$ for some $k\in\mathbb{R}$, or more generally, an exponential/logarithmic structure depending on the domain.

Decomposition of Solutions via Symmetry

For certain equations, such as $f(x) + f(-x) = 2f(0)$, the solution decomposes into even and odd parts. Let $g(x)$ be any odd function (that is, $g(-x) = -g(x)$) and $p$ a constant. Then $f(x) = f(0) + g(x)$ satisfies the equation, and the set of solutions is parameterized by all odd functions $g(x)$.

More generally, the use of substitution and symmetry arguments underpins the identification of particular classes of solutions, which may be further constrained by initial conditions or the requirement of compatibility with the function's domain and codomain.

Relation between Functional Equations and Types of Functions

The form of a functional equation often determines the type of function that can be its solution. For equations expressible through polynomials, logarithms, exponentials, reciprocals, or trigonometric forms, the solution space correlates with the algebraic or analytic properties of these standard functions. The graphical representation of these functions further illustrates the diversity of solution sets meeting the function equation requirements, as explored in various function types and visualized in depth in the Functions and Its Types resource.

Distinction from Algebraic and Differential Equations

Unlike algebraic equations, where the unknowns are numeric variables, or Differential Equations, wherein the unknowns are functions together with their derivatives, functional equations pose the problem of determining function structure directly from prescribed relationships between the values of the function at different inputs.

Conclusion

Functional equations are a fundamental aspect of higher mathematics and mathematical olympiad problem solving. The core techniques involve substitution, symmetry, properties of function types, and rigorous logical manipulation. Complete solution involves the full use of algebraic, analytic, and combinatorial tools, always taking care to examine the domain, codomain, and regularity conditions for admissible functional forms. Mastery of these topics forms a foundation for advanced mathematical reasoning and problem solving.