# 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).

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### Functional equation $4f(x^2+y^2)=(f(x)+f(y))^2$

Consider the following problem: Determine all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ that satisfy the functional equation $$4f(x^2+y^2)=(f(x)+f(y))^2.$$ So first of all plugging in $x=0=y$,...
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### Is there a less-trivial integer function with described properties?

To be found are integer one-valued functions $f(n_1,n_2)$ with following properties: $f(n_1,n_2)=f(n_2,n_1)$, $f(f(n_1,n_2),n_3)=f(n_1,f(n_2,n_3))$, $f(n_1+n_2,n_1+n_3)=n_1+f(n_2,n_3)$. So far I ...
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### If $B(x+y)-B(x)-B(y)\in\mathbb Z$ can we add an integer function to $B$ to make it additive?

Given a function $B:\mathbb R\to\mathbb R$ satisfying $B(x+y)-B(x)-B(y)\in\mathbb Z$ for all real numbers $x$ and $y$, is there a function $Z:\mathbb R\to\mathbb Z$ such that $B+Z$ is an additive ...
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### How to solve non-linear differential equation [duplicate]

How to solve non-linear differential equation $$y'(x) = y(y(x)), \quad y\colon\mathbb{R}\to\mathbb{R}?$$ Of course, $y(x)\not\equiv 0$. If we substitute $y(x) = Ax^n$, we get complex $n$ and $A$. Any ...
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### Functional Equation (no. of solutions): $f(x+y) + f(x-y) = 2f(x) + 2f(y)$

Find all the functions $f\colon\mathbb{Q} \to \mathbb{Q}$ such that $f(x+y) + f(x-y) = 2f(x) + 2f(y)$, for all rationals $x,y$.
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### $f(x^2) = 2f(x)$ and $f(x)$ continuous

I ran into a problem recently where I obtained the following constraint on a function. $$f(x^2) = 2f(x) \,\,\,\, \forall x \geq 0$$ and the function $f(x)$ is continuous. Can we conclude that $f(x)$ ...
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### If $f(x)^2=x+(x+1)f(x+2)$, what is $f(1)$?

Suppose $f$: $\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}$ and $f(x)^2 = x + (x+1)f(x+2)$, what is $f(1)$? Or more in general, what is $f(x)$? The motivation behind this problem is that I want to find ...
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### Solving $(f(x))^2 = f(\sqrt{2}x)$

I would like to know how to solve this equation : $$f(x)^2 = f(\sqrt{2}x)$$ We assume that $f : \mathbb R \to \mathbb R$ is $\mathcal C^{2}$. The answer should be $f(x)=e^{-x^{2}/2}$, but I don't ...
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### A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...
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### Functional equation: $R(1/x)/x^2 = R(x)$

The following can be shown without much hassle. Suppose $R$ is a rational function satisfying the following functional equation. \begin{align} \frac{1}{x^2} R\left( \frac{1}{x} \right) = R(x)...
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### Help remembering a Putnam Problem

I recall that there was a Putnam problem which went something like this: Find all real functions satisfying $$f(s^2+f(t)) = t+f(s)^2$$ for all $s,t \in \mathbb{R}$. There was a cool trick to ...
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### Prove that $f(x)=8$ for all natural numbers $x\ge{8}$

A function $f$ is such that $$f(a+b)=f(ab)$$ for all natural numbers $a,b\ge{4}$ and $f(8)=8$. Prove that $f(x)=8$ for all natural numbers $x\ge{8}$
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### References about Sierpinski's Theorem regarding Darboux functions

I am writing something about the following two theorems: Every function $f: \Bbb{R} \to \Bbb{R}$ can be written $f=f_1+f_2$ where $f_1,f_2:\Bbb{R} \to \Bbb{R}$ both have the Darboux property. ...
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### Functional equation of irreducible characters

I am preparing to an exam in representations of finite groups. I am trying to tackle a problem regarding a characterization of irreducible characters: Let $f$ be a complex-valued function on a finite ...
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### How to solve the functional equation $f(f(x))=ax^2+bx+c$

Find all real numbers $a,b,c\in\mathbb{R}$ for which there exists a function $f:\mathbb{R}\to\mathbb{R}$ such that: $$f(f(x))=ax^2+bx+c$$ for all $x\in\mathbb{R}$. The only thing I could deduce is: ...
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### A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The ...
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### Vector Space Structures over ($\mathbb{R}$,+)

Consider the abelian group ($\mathbb{R}$,+) of real numbers with the usual addition. Is there a scalar multiplication \cdot : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, \end{...
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let $f:R\longrightarrow R$, and $f$ is continous,and such that $f(f(x))=xf(x)+1$, find all this $f$? follow is my some idea:(but I don't have solution) We have $f(f(0)) = 1$, so there is your $c = ... 1answer 349 views ### Which positive continuous functions satisfy$F(x) = F(e^x)-F(e^{-x})$for$x\geq 0$? There is at least one such function. It is the cdf of the equilibrium probability distribution for the chaotic sequence$x(n+1) = |\log x(n)|$with$x(1) = 2$. Its graph (approximation) is pictured ... 1answer 273 views ### Find all functions on the non-zero reals to itself satisfying$f(xy)=f(x+y)(f(x)+f(y))$Find all functions$f\colon\mathbb{R}\setminus\{0\}\to\mathbb{R}\setminus\{0\}$such that$f(xy)=f(x+y)(f(x)+f(y))$. I'm reasonably confident that the solutions are$f(x)=\frac{1}{x}$and$f(x)=\frac{...
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The function $f:R_+\to R_+$ is continuously differentiable and increasing. Also, $f(0)=0$ and $f(\infty)=\infty$. Continuity and differentiability of higher orders can be assumed if necessary. ...
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### Solution(s) to $f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y))$

I would like to know the solutions of the functional equation: $$f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y)), \forall x,y\in\mathbb{R}$$ where $f:\mathbb{R}\rightarrow\mathbb{R}$. I have already determined ...
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### Non-trivial solutions of a functional equation $(f \circ f \circ f)(x)=x$

I have come to this from the topic of self-inverse functions. Let's consider a more complicated case where: $$(f \circ f \circ f)(x)=x \tag{1}$$ Also assume that $f(x)$ is continous on some non-...
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### Find all the functions $f: \mathbb{N} \to \mathbb{N}$ such that $(m+f(n))(n+f(m))$ is a perfect square for all $m,n$

Let $N$ be the set of natural number. Find all functions $f: \mathbb N \to \mathbb N$ , such that the number $(m+f(n))(n+f(m))$ is perfect square for all natural numbers $m$ and $n$. I was unable to ...
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