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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Correct math notation to show min and max of a list of numbers and sign?

I am working on a paper that uses a calculation from DIN 15018. The calculation is only described in text and not as a math equation. I would like to display as an actual equation if possible. The text from DIN 15018 (English version) is: The…
Andrew
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Solutions to functional equation $f(at+x)+g(x)+h(t,bt+x) =0 $

Let $a \neq 0$ and $b \neq 0$ be fixed constants with $a \neq b$. Find all twice continuously differentiable functions $f:\mathbb{R}\rightarrow \mathbb{R}$, $g:\mathbb{R}\rightarrow \mathbb{R}$ and $h:\mathbb{R}^2 \rightarrow \mathbb{R}$ such…
user103828
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Functional equation: $f$ is periodic with every irrational numbers as period

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number $r$ and any real number $x$ we have $f(x)=f(x+r)$. Show that $f$ is a constant function. It's easy to see any constant function satisfies the original property. But I…
Pikachuu.
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Trying to solve the equation involving floor function

I am trying to solve the following equation: Floor[logx+1]+x=11 Where Floor function returns the greatest integer smaller than the value in bracket. e.g. Floor[3.3] = 3 And the logarithm is to the base 10 I tried using WolframAlpha, but it's timing…
Sid
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Does a solution to this functional equation exist and if so can we construct it?

For $x\geq 0 $ we have $f(x) +xf(1/x) = x/(1+x)$ as well as the conditions $\lim_{x\rightarrow 0} f(x) = 0$ and $\lim_{x\rightarrow \infty} f(x) = 0$. Clearly $f(1) = \frac{1}{4}$. What is the solution, if it exists at all? I have no clue how to…
antarcticfox
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continuous functions on rationals

Let $\Bbb Q$ be the set of rationals and $f:\Bbb Q\to \Bbb R$ be a continuous function. Then $f$ is bounded on some interval? If not, what happen if in addition $f$ satisfies $f(xy)=f(x)f(y)$ for all $x, y\in \Bbb Q$?
Chung. J
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Satisfying Functional Equations

Find all functions ${\rm f}:{\mathbb N}\times{\mathbb N} \rightarrow {\mathbb N}$ satisfying $$ \begin{array}{rrcl} a) & {\rm f}\left(n,n\right) & = & n \\[2mm] b) & {\rm f}\left(n,m\right) & = & {\rm f}\left(m,n\right) \quad\forall\ m,n\ \in…
r9m
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Finding all functions on $ \mathbb R $ satisfying $ f ( x - | x | ) + f ( x + | x | ) = x $

I'm not sure if I'm correct in this example: Find all functions such that $ f ( x - | x | ) + f ( x + | x | ) = x $, where $ x \in \mathbb R $. So my answer is the are only one function satisfying this condition $ f ( x ) = \frac x 2 $. Am I…
Mark
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function equation with translation of independent variable: $\frac{f(x+a)}{f(a)}=g(x)$

The following has come up in some work I'm doing: If $\frac{f(x+a)}{f(a)}=g(x)$, where $g(x)$ is given and $a \ge 0$ is a constant, what is $f(x)$? We can assume that $g(x)>0 \ \forall x$ . Of course a solution would be great, but I'd appreciate…
Greg Reese
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Help with the functional equation $f(x - f(y)) = f(f(y)) + 2 x f(y) + f(x) - 1$

I am asked to find all $f : \mathbb{R} \to \mathbb{R}$ such that $f(x - f(y)) = f(f(y)) + 2 x f(y) + f(x) - 1$, which I solved and got that $f(x) = 1 - x^2$ -- a correct solution and everything. What puzzles me is how do I find all solutions. I…
d125q
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Solving the functional equation $g(x)=g(a^mx)c^m$

Let $a\in\mathbb{C}^*$ with $|a|\not=1$. Let $m\in\mathbb{Z}$. Find all functions $g:\mathbb{C}^*\to\mathbb{C}^*$ and constants $c\in\mathbb{C}^*$ such that $g(x)=g(a^mx)c^m$. I know one possibility is to let $n\in\mathbb{Z}$ then $c=a^n$ and…
user44322
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Functional equation $f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$

Find all of the functions defined on the set of integers and receiving the integers value, satisfying the condition: $$f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$$ for each pair of integers $(a,b)$.
Mark
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solving a functional equation $f(xy)=f(x)+f(y)-1$ on positive integers using given values

Moderator Note: This is a current contest question on Brilliant.org. The current contest ends on 13 October 2013, after which time this question will be unlocked. A Function $f$ from the positive integers to the positive integers satisfies the…
tattwamasi amrutam
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The functional equation $ f ( m n ) = f ( m ) f ( n ) $ does not have a strictly increasing solution with $ f ( 2 ) = 3 $.

Show that there does not exist a function $ f : \mathbb N \to \mathbb N $ which satisfy $ f ( 2 ) = 3 $; $ f ( m n ) = f ( m ) f ( n ) $ for all $ m , n \in \mathbb N $; $ f ( m ) < f ( n ) $ whenever $ m < n $.
priya
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Finding a polynomial $P(x)$ with real coefficients satisfying $P(x)P(x+1)=P(x^2+2)$

$P(x)$ is a polynomial with real coefficients satisfying $$P(x)P(x+1)=P(x^2+2)$$ I did find some solutions for $P(x)P(x+1)=P(x^2)$ and I wonder if I can do the same for this problem I tried making $P(x)=a_nx^n+\cdots+a_1x+a_0$, and got that $a_n=1$,…
Kii
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