Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation(s) and possibly other conditions; e.g., the goal can be to find all continuous solutions. Solving the equation means finding all functions satisfying the given equation(s) and any additional conditions.This is different from the more common use of the word "equation", where the solutions are numbers. It is also different from the more common use of the word "functional", referring to a mapping from a space into the reals or complexes. For basic questions about functions use more suitable tags like or .

A common technique used in solving functional equations is finding some properties of satisfying functions by substituting variables for certain values in the equation. Proving properties of satisfying functions is also helpful - finding that a function is injective, surjective, involutive, and so on, is often a key step in finding all possible solutions. Other techniques such as exploiting symmetry, considering fixed points, and even using certain properties of domains (e.g. well-ordering) sometimes help.

Some well-known functional equations are:

More information can be found at Wikipedia.

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Checking my understanding of Cauchy's functional equation $f(x+y)=f(x)+f(y)$

Cauchy's functional equation is given as $$f(x+y)=f(x)+f(y)$$ Wikipedia states that the solution to this functional equation with $x\in\mathbb Q$ is $f(x)=cx$, where $c$ is an "arbitrary rational" number. Without adding more constraints, one…
Simply Beautiful Art
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The functions satisfying $ \psi ( x + 2 ) = 1 + \sqrt{ 2 \psi ( x ) - \psi ( x ) ^ 2 } $

The function $ \psi : \mathbb R \to \mathbb R $ satisfies the relation: $$ \psi ( x + 2 ) = 1 + \sqrt{ 2 \psi ( x ) - \psi ( x ) ^ 2 } \text , $$ for all real $ x $. What properties does such function have? Give an example of of such function that…
Roman83
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If $g:\mathbb{N}\rightarrow \mathbb{R}$ and $g(m+n)+g(m-n)=2g(m)+2g(n)$ what is $g(x)$

Determine all functions $g:\mathbb{N}\rightarrow \mathbb{R}$ such that $g(1)=1$ and $$g(m+n)+g(m-n)=2g(m)+2g(n), \quad \forall m\ge n \in \mathbb{N}$$ Because of the identity $k\cdot (a+b)^2 +k\cdot (a-b)^2=2k\cdot a^2+2k\cdot b^2$ I guess it is…
user302454
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Simple almost linear functional equation

I'd like to solve functional equation: $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}.$ I've managed to get: $f(0)=0,f(n)=0$ for all $n\in N$; $f(\frac{1}{2})=0$; $f(-x)=-f(x)$. I'll be grateful for any help.
leg14able
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Find all continuous functions $f$ over real numbers such that $f(x)+x = 2f(2x)$

Find all continuous functions $f$ over real numbers such that $f(x)+x = 2f(2x)$. We have $f(0) = 0$ and $f(x) = 2f(2x) - x$, but I am not sure how to convert this functional equation into something that is easier to solve. Maybe using induction…
user19405892
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Functional Equation $f(f(x))=af(x)+bx$

For real numbers $a$ and $b \neq 0$, $f(x)$ satisfies the following: $$f(f(x))=af(x)+bx$$ (1) $f(x)$ is continuous and $00>b, a^2+b \leq 0$, show…
zxcvber
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Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$

Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$. We know that $f(0) = 0$. Now set $x = n$ and $y=n$ to get…
user19405892
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Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$:$f(f(f(x)+y)+y)=x+y+f(y)$

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$: $f(f(f(x)+y)+y)=x+y+f(y)$ I got the following: (1)$f$ is injective (2) $f(0)=0$ (3)$f(f(f(x)))=x$ But then how to proceed?
user321656
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Question about solutions of nonlinear functional equation $f(x)^2=xf(2x)$

One of the basic nonlinear functional equations is the following one: $$f(x)^2=xf(2x),\quad x>0\text.$$ I found out that functions $f(x)=2^{1-x}x\exp(cx)$ form the family of solutions of this equation. But do this family cover all possible solutions…
Peter95
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Solve functional equation $f(x_1 x_2) = g_1(x_1) g_2(x_2)$

Let $x_1$ and $x_2$ be real positive numbers. The problem is to find all possible triples of $f$,$g_1$,$g_2$ such that $f(x_1 x_2) = g_1(x_1) g_2(x_2)$. I suspect that the only one solution is $f(x_1 x_2) = (x_1 x_2)^n$, $g_1(x_1) = x_1^n$,…
0x2207
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Why does the monotonicity imply $2^u < 3^v$ if and only if $3^u < 6^v$?

In the question and solution below, I am wondering how to #$7$ it says "The monotonicity of $f$" implies that $2^u < 3^v$ if and only if $3^u < 6^v$, $u,v$ being positive integers." How does this even depend on the definition of $f$? And if it does…
Jacob Willis
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Functional equation $f(x+y)=\frac{f(x)+f(y)}{1-4f(x)f(y)}$ with $f'(1)=1/2$

Try to find the solution of the functional equation $$f(x+y)=\frac{f(x)+f(y)}{1-4f(x)f(y)}$$ with $f'(1)=1/2$.
Pippo
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Do I need to verify solutions to functional equations?

I am studying the (basics of) solving functional equations. My teacher stipulates that we check any solutions obtained by substitution. Similar guidelines are given in this IMO training material. For a typical example, consider $f:\mathbb{R}…
Minethlos
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Functional Equation similar to Cauchy's: $\{f(x)\}+\{f(y)\}=\{f(x+y)\}$

Find all functions $f:\mathbb{Q}\to \mathbb{Q}$ such that for any $x,y\in{}\mathbb{Q}$ we have $$\{f(x)\}+\{f(y)\}=\{f(x+y)\}\text.$$ Note that $\{t\}$ denotes the fractional part of $t$ for instance $\{1.5\}=0.5$ Progress: Since $t-\lfloor…
Jack Frost
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Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^{2}$ . Then $f(3)$ =?

Options: (a)$4$, (b)$4f(0)$, (c)$4-f(0)$, (d)$4+f(0)$, (e)$16+f(0)$. CORRECT ANSWER USING REDUCTION Deep thanks to @martini and @A.S. , soo much respect . Since we don't exactly know the nature of $f(x)$ we will start of by…
Sujith Sizon
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