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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Function preserving exponentiation

I'm wondering what kind of function preserves exponentiation, i.e., what is an $f$ such that $f(a^b)=f(a)^{f(b)}$?
nullgraph
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Solutions for the Functional Equation $f(x^2)=f(x)^2$

Suppose a continuous function $f:[0,1]\rightarrow [{0,1}]$ satisfies the functional equation $f(x^2)=f(x)^2$. Then I conjecture that we must have $f(x)=0$ or $f(x)=x^r$ for some real number $r\geq 0$. However, I haven't the foggiest idea how to…
dcw
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Functional Equation - Am I right?

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$ So here's my solution, If $x=y=0$, $2f(0)=2f(0)^2$ $\implies f(0)=0$ or $f(0)=1$. Case $1$: $f(0)=0$ If…
John Marty
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An elementary functional equation: $f\big(x+f(x)\big)=f(x)$

I am finding this functional equation from a past high school mathematics competition rather tricky. Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}^+$ such that: $f\big(x+f(x)\big)=f(x)\quad \forall…
goonfiend
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functional equation $f(x^2)=xf(x)$

This must be widely known, but it is not to me. What is the solution of the functional equation $f(x^2)=xf(x),\,\forall x\in\mathbb R$ or for $x$ in a finite field? What is it when $f$ is continuous or differentiable when $x\in\mathbb R$? Obviously…
Hans
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How find the function $f(f(x)+xy)=f(x)+xf(y),\forall x,y\in R$

Find all function $f:R\to R$ and such $$f(f(x)+xy)=f(x)+xf(y),\forall x,y\in R$$ Let $x=y=0$,then $$f(f(0))=f(0)$$
math110
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I am learning functional equation and encountered $f(f(x)-x)=x$.

I was self-learning functional equations because I will be learning functional equation in Math Team soon. And I made up the following problem: Find all solutions to functional equation $f:\mathbb{R}\rightarrow\mathbb{R}$ $$f(f(x)-x)=x$$ I tried…
Culver Kwan
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Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(f(x)+yz)=x+f(y)f(z)$

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(f(x)+yz)=x+f(y)f(z)$ I was told to do this by proving $f$ is injective and surjective. I have proved it this way : setting $y=z=0 $, and then…
LOIS
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How to solve equations in the form $af^2(x)+bf(x)+cx=0$?

I have not seen any books about solving the equation of the following form: $af^2(x)+bf(x)+cx=0$ where $a$, $b$, $c$ are constants and $f^2(x)=f(f(x))$. We are going to find an expression of the function $f$. If I substitute $f(x)=kx$ in, I can get…
Ma Joad
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Could one define a function in the following way, without leading to a contradiction?

I was pondering, could one define a function such that $$T(a,b)^{T(c,d)}=T(a^c,b^d)$$ $∀a,b,c,d$? I asked the same question some minutes ago, and someone was able to find an example, that of $T(x,y)=1, ∀x,y$. And as much as this is completely…
Sam
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$f(xf(y)-f(x))=2f(x)+xy$

This is my problem "Find $f:\mathbb{R}\to \mathbb{R}$ s.t. $f(xf(y)-f(x))=2f(x)+xy$ ". I'm not so good at functional equation so all I did until now is subtituting $x,y$ by $1,0$ and I couldn't even calculate a value. One other approach is that I…
user399246
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Functional equation: $f(x)f(y)$ is radially symmetric

Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that $f(x) f(y)$ is radially symmetric, i.e. $$f(x) f(y) = g \left( \sqrt{x^2 + y^2} \right)$$ for some function $g : [0, \infty) \to \mathbb{R}$. (Note: I've come up with my…
Daniel Schepler
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6 answers

Solving functional equation $f(x+y)=f(x)+f(y)+xy$

We are given $f(0)=0$. Then when $x+y=0$: $$0=f(-y)+f(-x)+xy$$ Can I now use $x=0$ and obtain: $$0=f(-y)?$$ Is this correct? Is there a better way to solve this equation?
user263286
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Functional equation $f(x+y)-f(x)-f(y)=\alpha\big(f(xy)-f(x)f(y)\big)$ is solvable without regularity conditions

I was reviewing this question and got motivated to solve this general problem: Find all functions $f:\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, $$f(x+y)-f(x)-f(y)=\alpha\big(f(xy)-f(x)f(y)\big)$$ where $\alpha$ is a nonzero…
Mohsen Shahriari
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Functional equation $f(x+y)+f(x-y)=2f(x)f(y)$

Let $f:\mathbb{R}\to\mathbb{R^*}$ be a function such that $f(x+y)+f(x-y)=2f(x)f(y),\forall x,y\in\mathbb{R}$. Prove that $f(x)=1,\forall x\in\mathbb{R}$. I have managed to prove the following: 1) $f(0)=1$ Set $x=y=0$ so: $f(0)=f^2(0)\Rightarrow…
user171110
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