# 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).

2,516 questions
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### Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ so that $f(x)f(y)- \frac{4}{9} xy= f(\!x+ y\!)\,(\!\forall x,\,y\in \mathbb{R}\!)$ .

Problem. Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f(x)f(y)- \dfrac{4}{9}\,xy= f(x+ y)\,\,(\!\forall x,\,y\in \mathbb{R}\!)$ (1). My above problem given a solution, and I'm ...
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### Functional equation for tan

If $f$ is a differentiable function on $\mathbb{R}$ and $f'(0)=2$ satisfying $$f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)},$$ then to prove that $f(x)=\tan 2x$. I know that we must prove using the first ...
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### Find all functions for $f:\Bbb{N}\to\Bbb{N}$ such that $f\left(m^2+f(n)\right)=f\left(m^2\right) +n$

I would have given my approach but i didnt get anywhere. I just substituted zeroes and got $f(f(n)) =n$ and I'm just lost. Any help would be appreciated
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### $f:\mathbb{R_{\geq 0}} \to \mathbb{R_{\geq 0}}$ such that for all $x$ we have $xf(1+xf(y))=f(f(x)+f(y))$

Find all nonnegative real number $a$, such that $f(a)=0$ for any function $f$ satisfying: $xf(1+xf(y))=f(f(x)+f(y))$ with all $x,y$ are nonnegative real number. I don't know why this problem only ...
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### Functions $f$ that $f(f(x))=x$, but $f:S^1\to S^1$

Background Denote $e_A$ the identity map from $A$ to itself. Questions such like solving $f$ in the functional equation $f\circ f=e_\mathbb{R}$ or $f\circ f=e_{\mathbb{R}\setminus\{a_1,\ldots,a_n\}}$ ...
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### A problem with functions defined on positive integers.

Where [x] denotes the greatest integer number, which does not exceed x. I need some help please. The proof should also be at high school level. Please don’t use hard or complex things.
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### Finding all positive real functions satisfying $xf(y)+f(f(y))\leq f(x+y)$

Find function $f: \mathbb{R}_{> 0}\rightarrow \mathbb{R}_{> 0}$ such that: $xf(y)+f(f(y))\leq f(x+y)$ for all positive $x$ and $y$? That problem made me think a lot. This is the first time I ...
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### $f(e^x) = e^{f(x)}$, what is f? [duplicate]

Find all functions $f$ and their domains, such that $f(e^x) = e^{f(x)}$ I have verified that the functions below satisfy the equation for certain domains. Would these be the only solutions? But how ...
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### $f(ax)=f(x)^2-1$, what is $f$?

Suppose $f(ax)=(f(x))^2-1$ and suppose that $f$ is analytic in some neighborhood of $x=0$. Expanding in power series, we get $a=1+\sqrt{5}$ or $1-\sqrt{5}$. We take positive $a$. If $f\neq{\rm const}$ ...
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### Solve the functional equation $f(x+1)-f(x)=x*\sin(x)$ [closed]

Solve $f(x+1)-f(x)=x*\sin(x)$
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### Rational Functional Equations

Suppose $f(x)$ is a rational function such that $3 f \left( \frac{1}{x} \right) + \frac{2f(x)}{x} = x^2$ for all $x \neq 0$. Find $f(-2)$. I tried substituting different values of $x$ to get a system ...
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### which is the function of (3) based in this equation

If we know that $(f\circ f)(x)=4x+3$, with $f(0)=4$, what is $f(3)=?$ I have found that $f(f(x))= 16x+15$, but I don't know where to go from there.
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### I can compute a function $F(x)$ such that $F(x(1/2-x))= F(x)/2$, It is analytic on a filled Julia set.

I have been studying a function $F(x)$ obeying $F(p(x))=F(x)/2$. I did numerical work for $p(x)=x(1/2-x)$, although a similar functional equation could be solved for any polynomial with an attractive ...
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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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### Uniqueness of the solution of non-linear ODE of second order

Let $n$ be an integer with $n>3$ and $f \colon [0,\infty ) \to \mathbb{R}$ be a solution of $t^{1-n}(t^{n-1}f'(t))'=f(t)|f(t)|^{\frac{4}{n-2}}$ with initial values $f(0)=a$ and $f'(0)=0$. Then, is ...
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### Restricted Cauchy equation on a non-dense domain

It is really well known that if $f: \mathbf{R}\to \mathbf{R}$ is continuous and $$\forall x,y \in \mathbf{R},\,\,\,\,f(x+y)=f(x)+f(y)$$ then $f$ is linear, i.e., there exists $a \in \mathbf{R}$ such ...
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### How many functions can be defined from natural number to natural number such that LCM(f(n),n) -HCF(f(n),n) is less than 5.

How many functions $f:\mathbb{N}\to\mathbb{N}$ can be defined from natural number to natural number such that $$\text{LCM}(f(n),n) -\text{HCF}(f(n),n) <5$$ LCM is least common multiple HCF is ...
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### What is a function $f$ such that $f(f(x))=2f(x)$ and $f(0)=1$?

Is there a function $f$ such that $f(f(x))=2f(x)$ and $f(0)=1$? I don't know how to attack this problem. How do I solve an equation for a function?
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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 ...
Be $f:\mathbb R^ +\mapsto\mathbb R$ a function that satisfies the following conditions: a)$f(f(f(x)))+2x=f(3x)$ for every $x\gt 0$; b) $\lim_{x \to \infty} (f(x)-x)=0$. This was proposed by ...