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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Functions over $\mathbb C$ that are both additive and multiplicative

If we have a function $f:\mathbb R\to\mathbb R$ so that, for all reals, $$f(x)+f(y)=f(x+y)\text{ and }f(x)f(y)=f(xy),$$ we can conclude that it is either identically $0$ by the following argument: We have $f(x)^2=f(x^2)$, so nonnegative $x$ are…
Carl Schildkraut
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functional equations

Find all solutions $f:\Bbb R^2 \to \Bbb R$ satisfying $$ f(xu-yv, yu+xv)=f(x, y)f(u, v). $$ Solution of the following equation $$ f(xu+yv, yu-xv)=f(x, y)f(u, v) $$ is known as $$ f(x,y)=m(x^2+y^2), $$ where $m$ is multiplicative function on $\Bbb…
Chung. J
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Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x + y) = x + f(y)$

I'm engaging in the quest for understanding functional equations and I am trying to solve the problem: Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x + y) = x + f(y)$ This is what I have done so far: Let $y = 0$, then: $f(x +…
René Quijada
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What conditions must apply for functions $f$ and $g,$ so that $f(g) = g(f)?$

I have found some solutions, but nothing general. The solutions are $f=g; f = g^{-1}; f(x)=x; f(x)=ax, g(x)=bx,$ for any $a$ and $b;$ I am unable to find any solutions other than these, or to prove that none exist.
Karen Mossoyan
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Solutions to the functional equations $f(sx) = tx$ and $f(sx + (1-s)) = tf(x) + (1-t)$ on $[0,1]$

Suppose that $s,t \in (\frac{1}{2},1)$ with $t \ne s$. Does there exist a continuous bijection $f \colon [0,1] \to [0,1]$ which simultaneously satisfies the functional equations $$ f(sx) = tx $$ and $$ f(sx + (1-s)) = tf(x) + (1-t) $$ for all $x \in…
Zorngo
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What is the solution $f:\mathbb{R^3}\to \mathbb{C}$ to the functional equation $f(A)f(B)=f(A+B+A\times B)$?

Given $3$-vectors $A$ and $B$ and an unknown function $f:\mathbb{R^3}\to \mathbb{C}$, are there any solutions to this functional equation? $$f(A)f(B)=f(A+B+A\times B)$$ apart from the trivial solutions $f(A)=1$ or $f(A)=0$. Edit: As a simpler…
zooby
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Functional equations [Sample paper of Indian Mathematical Olympiad]

Edit- There was information missing (lack of clear printing in my book) in the book through which I referred the question. Confirming with my friend's book I have made a small change. I am really sorry. Edit 2- Guys, this question is meant for an…
Jayant Jha
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How to find the solutions of this functional equation

$$f(\tfrac{1}{2}+x)+f(\tfrac{1}{2}-x)=8xf\big(4(\tfrac{1}{2}+x)(\tfrac{1}{2}-x)\big)\qquad\text{for}\qquad x\in(0,\tfrac{1}{2})$$ I have no idea about how to tackle this equation. The original problem asks me to verify that a function indeed fits…
Apocalypse
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$h(x,y)=f(x)+g(y)$

As a followup comment to my answer to the question $\;\;\;\;$Can I split $\frac{1}{a-b}$ into the form $f(a)+f(b)$? "Lord Shark the Unknown" made the following observation: If $h:\mathbb{R}^2\to\mathbb{R}$ and $f,g: \mathbb{R}\to\mathbb{R}$ are…
quasi
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$f(x,y) - f(x,z) = g(y,z)$ implies $f(x,y) = a(x) + b(y)$

A result I need is: If $f(x,y) - f(x,z) = g(y,z)$ for all $(x,y,z)$, then $f(x,y) = a(x) + b(y)$ for some functions $(a,b)$. This seems almost obvious, and I've constructed a proof, but that proof seems unnecessarily complicated and is remarkably…
Ethan Ligon
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Solving tricky functional equation resembling quadratic equation

I have the following functional equation in hand, I can easily solve it for the case $(a, b ,c)=(1, 1,0)$ which gives $f(x)$ to be $x^2+x$. $\begin{aligned}{g(x)=a\left[f(x)\right]^2+bf(x)+c, \text{where } g(x)=x^4+2x^3+2x^2+x}\end{aligned}$ I…
Paras Khosla
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An interesting functional equation

$$\frac{1-f\Big(\frac{x}{x+(1-x)f(x)}\Big)}{1-f(x)} = 1-x(1-x)\frac{f'(x)}{f(x)}$$ Now, we know that $f(x)=c$ and $f(x) = \frac{a+bx}{1-x}$ are two solutions. How can I get other solutions or to prove that these are all the solutions? Thanks!
ftor
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find all fucntions such that $f(x+y) \geq f(x) + yf(f(x)) $

Find all functions $f:\mathbb{R}_+ \to \mathbb{R}_+$ (not necessarily continues function) where $\mathbb{R}_+ = ${$r \in \mathbb{R} : r \geq 0$}, such that $$f(x+y) \geq f(x) + y f(f(x)) \quad\forall x,y \in \mathbb{R}_+$$ I tend to believe that…
ned grekerzberg
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A doubt in the solution of IMO 1977 B3 question

The following question appeared in IMO 1977: "Let $f(n)$ be a function defined on the set of all positive integers and with all its values in the same set. Prove that if $$f(n+1)>f[f(n)]$$ for each positive integer n then prove that $$f(n)=n$$ for…
saisanjeev
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What's the solution of the functional equation $f(g(x)+y) = g(f(x)+y)$?

I need help with this: Find all functions $f, g : \mathbb{Z} \to \mathbb{Z}$, with $g$ injective and such that: $$f(g(x)+y) = g(f(x)+y), \text{ for all } x, y \in \mathbb{Z}\text.$$
user62189
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