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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Solve the functional equation $f(x+a+f(y))=f(f(x))+a+y$

So let $a$ be a real number. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $f(x+a+f(y))=f(f(x))+a+y$, for all real $x,y$.
CryoDrakon
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Find all functions $ \mathbb{Z} \mapsto \mathbb{Z} $ such that $f(a-b+f(b))=f(a)+f(b)$

I found this problem in an old book of math, here is my try: $$P(a;a) \mapsto ff(a)=2f(a)$$ $$P(a;f(b)) \mapsto f(a+f(b))=f(a)+2f(b)$$ $$P(0;b) \mapsto f(0)=0$$ $$P(0;b) \mapsto f(f(b)-b)=f(b)...(1)$$ $$P(b-f(b);b)\mapsto f(b-f(b))=-f(b)..(2)$$ From…
Maxwol
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Is operator $T^2=T$ with closed image and kernel bounded?

Let $X$ be a Banach space and let $T:X \rightarrow X$ be a linear map such that $T^2=T$ and both $\text{Im}(T)$ and $\text{Ker}(T)$ are closed. Then $T$ is bounded? I got stuck in this problem while I was studying functional analysis. Can anyone…
Silement
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Curious equation I came up with

I was just deadly curious about a solution of a certain functional equation, or better, a question. Given $f(x)$ and $g(x)$ as two functions domained in $A$ and $B$, and given the equation for f(x) $$f(x)=g(f(x))$$ I found out, with some weird…
us er
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let $f( x + \frac{ 1}{ x } ) = x ^ 2 + \frac{ 1}{ x ^ 2} $ then $f(x)$ equals

let $f( x + \frac{ 1}{ x } ) = x ^ 2 + \frac{ 1}{ x ^ 2} $ then $f(x)$ equals options are $( A ) x ^ 2 - 2$ $( B ) x ^ 2 - 1$ $( C ) x ^ 2$ I don't know how to solve this type of problems
Mohit
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How to solve the functional equation $ f(x^2+xf(y))= xf(y)$

Hello please how to find all the functions $f:\mathbb{R}\to \mathbb{R}$ such that $$ f(x^2+xf(y))= xf(y)$$ I see that $f(0)=0$ but how to do after
Vrouvrou
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Functional Equations and information theory: $f(xy)=f(x)f(y)$ for all $x,y\in[0,1]$

What is the solution of the functional equation $f(xy)=f(x)f(y)$ for all $x$ and $y$ in $I$ where $f$ is a real-valued mapping with domain the unit closed interval $I$?
Kitu
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Find all functions $f:\Bbb R \rightarrow \Bbb R$ such that for every $x,y \in \Bbb R$: i) $f(x)\leq x$ ii) $f(x+y)\leq f(x)+f(y) $

Find all functions $f:\Bbb R \rightarrow \Bbb R$ such that for every $x,y \in \Bbb R$: i) $f(x)\leq x$ ii) $f(x+y)\leq f(x)+f(y) $ In functional equations, I have trouble when inequality has a role...
Hamid Reza Ebrahimi
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What can be $f$ so that $f(f(x)) = -x$?

What can be $f$ so that $f^2(x) = -x$ for all $x\in R$? I know that if $f^2(x) = -x$ then $f(x)$ is injective and $f$ can not be continuous. But I can not find an example of discontinuous function so that $f^2(x) = -x$ for all $x\in R$. Can anyone…
anonymous
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How to find the derivative of function using given functional equation?

$$f \left(\frac{x+y}{2} \right) = \frac{f(x)+f(y)}{2}, \forall x,y \in \mathbb{R}$$ $$f'(0)= -1,\space f(0)=1$$ $$f'(u)=?$$
Aakif Akhtar
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Solving for this function

From the Indian National Mathematics Olympiad 1992: Determine all functions $f: \mathbb R -[0,1] \rightarrow \mathbb R$, satisfying the functional relation: $$f(x) + f \left( \frac{1}{1-x} \right) = 2\frac{1 - 2x}{x(1-x)}$$ where $x$ is a real…
buzaku
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$g(x+y)=g(x)g(y)$ for all $x,y \in\mathbb{R}$. If $g$ is continuous at $0$, prove that $g$ is continuous on $\mathbb{R}$

$g(x+y)=g(x)g(y)$ for all $x,y \in\mathbb{R}$. If $g$ is continuous at $0$, prove that $g$ is continuous on $\mathbb{R}$.
UNM
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2 answers

Find continuous $f$ that satisfies $f(0)=1$, $f(m+n+1)=f(m)+f(n)$

Let $f$ be a continuous real-to-real function that satisfies $f(0)=1$ and $$f(m+n+1)=f(m)+f(n)$$ for all $m,n\in{\mathbb R}$. Show $f (x) = x + 1$ for all $x$. What is the solution? Source: Solving mathematical problems a personal perspective,…
MENZIES
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Functional equation $f(ax)=bf(x)$

What are all the solutions to the functional equation $f(ax)=bf(x)$, where $a,b>0$, and $f$ is continuous, strictly monotone and increasing, and $x$ ranges over the reals? references? proof? Thanks Additional details following the first…
user154729
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Find all the $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f\left(f^3(x)+y^3\right)=x^2+f^3(y)$.

Find all the $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f\left(f^3(x)+y^3\right)=x^2+f^3(y)$, where $f^3(x)$ stands for $[f(x)]^3$. I really don't know where to start off.
GraphBuster
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