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.

3976 questions
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Help with functional equation $F(x,x)+F(x,c-x)-F(c-x,x)-F(c-x,c-x)=0$

How can we find $F$ satisfying: exists a $c$ such that $$F(x,x)+F(x,c-x)-F(c-x,x)-F(c-x,c-x)=0 \text{ for all } x,y $$ Several quadratic polynomials in $x,y$ satisfy the above property. I'm trying to generate more examples of functions that satisfy…
Sergio Parreiras
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Finding all $ f : \mathbb Z \to \mathbb Z $ that satisfy $ f ( 0 ) = 1 $, $ f \big( f ( x ) \big) = x $ and $ f \big( f ( x + 1 ) + 1 \big) = x $

Find all functions $ f : \mathbb Z \to \mathbb Z $ that satisfy the following conditions: (i) $ f ( 0 ) = 1 $; (ii) $ f \big( f ( x ) \big) = x $ for all integers $ x $; (iii) $ f \big( f ( x + 1 ) + 1 \big) = x $ for all integers $ x $. How can I…
Denise
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Solve the functional equation, $f(x+y+1)= \left(\sqrt{f(x)} + \sqrt{f(y)}\right)^2$

Solve the functional equation, find all real valued functions $f(x)$ s.t. $$f(x+y+1)= \left(\sqrt{f(x)} + \sqrt{f(y)}\right)^2$$ given $f(0)=1$. I reached to a point that $4f(x)=f(2x+1)$
maths lover
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Multiplicative function on rationals

Let $\Bbb Q^+$ be the set of positive rational numbers. Find all solutions $f:\Bbb Q^+ \to \Bbb R$ of the functional equation $$ f(xy)=f(x)f(y), \quad x, y\in \Bbb Q. $$ Is $f(x)=x^a$ the only solution? If not, is it true if we assume that $f$ is…
Chung. J
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Solve functional equation

find all functions $f:\mathbb{R^{*}}\to \mathbb{R}$ such that $$f(y^2f(x)+x^2+y)=x(f(y))^2+f(y)+x^2,\;\forall x,y\in \mathbb{R^{*}}$$ ($\mathbb{R^{*}}=\{x\in\mathbb{R},x\ne 0\})$
Babymath
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Solving the functional equation $f\left(y^2f(x)+x^2f(y)\right)=xy\big(f(x)+f(y)\big)$

Problem: find all continuous functions $f:[0,+\infty)\to [0,+\infty)$ such that $$f\left(y^2f(x)+x^2f(y)\right)=xy\big(f(x)+f(y)\big),\;\forall x,y\in [0,+\infty)\text.$$
Babymath
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Solutions to $g(ab) = ag(b) + bg(a)$ - "Zero function question"

This question was recently asked and then voluntarily removed by its author. I thought it was interesting enough to get an answer. The question was as follows: "The function $g:\mathbb R\to\mathbb R$ satisfies $g(ab)=ag(b)+bg(a)$" For this…
Chris Culter
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Can a non-constant function satisfy $f\left(\frac{x+y+1}{2}\right) = \frac{f(x)+f(y)}{2}$?

I'm trying to find functions $f$ over $\mathbb R$ which satisfy $$ f\left(\frac{x+y+1}{2}\right) = \frac{f(x)+f(y)}{2} $$ for all real $x$ and $y$. One thing I immediately note is that shifting any potential function by a constant preserves the…
Randall
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Solution or property to a functional equation

I have a continuous and nondecreasing function $F:\mathbb{R}\to\mathbb{R}$ satisfying following conditions: $F(x) = 0$ for any $x\in (-\infty, 1]$, $\lim\limits_{x\to +\infty} F(x) = 1$, for any $x\in \mathbb{R}$ we have $$F(x) \;=\; 0.8 F(2x - 1)…
Greenhand
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Weird Functional Equation problem on the irrationals

This weird question was given by my professor as a part of my assignment : $\textbf{Question :}$ “Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that it satisfies $f(x + y) = x f(\frac{1}{y}) + y f(\frac{1}{x})$, whenever $x$ and $y$ are both…
Vinay Karthik
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Find all polynomials $P(x)$ with real coefficients satisfying $P(x-1)P(x+1)=P(P(x))$

Find all polynomials $P(x)$ with real coefficients satisfying $P(x-1)P(x+1)=P(P(x))$ I tried $P(x) = c$ (const) and got $c=0$ or $c=1$. But when I tried $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}$, normally I would be able to find $a_{n}$ but in…
Kii
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Alternate solution to an IMO 1993 problem

Let $\mathbb N=\{1,2,3,…\}$. Determine if there exists a strictly increasing function $f:\mathbb N→\mathbb N$ such that $f(1)=2$ $f(f(n))=f(n)+n$ for all $n$. Recently I posted the following solution I came up with We can see $…
Aarush Saharan
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IMO problem from 1993 on functional equations

Let N={1,2,3,…}. Determine if there exists a strictly increasing function $f:N→N$ such that $f(1)=2$ $f(f(n))=f(n)+n$ for all n." Here is the solution I came up with We can see $f(1)=2$ $f(2)=3$ $f(3)=5$ Basically if $a_n$ is the $(n+2)^{th}$…
Aarush Saharan
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Is there a method to solve the equation $f( f(2x) - f(x) ) = x$?

I've already derived it, replaced it with other variables, applied the function in the form of a power series, but we always end up with an even more complex problem. With a simple look we can see that $f(x) = \pm x$ is a trivial solution, but how…
JaberMac
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Find all $f(x)$ such that $f(4x) + 2f(x+1) = \sqrt{x} + \sqrt{x+1}$, for all $x \ge 0$.

A particular solution is: $f(x) = \frac{1}{2} \sqrt{x}$, obs: don't use this as a given, this is just to show that there is at least one solution. What method should i use to solve this type of equation?
JaberMac
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