# 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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### Find all functions satisfy an equality

The question: Find all functions $f$ defined over $\mathbb{R}$ satisfying the equality: $\forall x,y \in \mathbb{R}$ $$f(y - f(x)) = f(x^{2002} - y) - 2001y f(x)$$ How do I approach (any hints) to ...
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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 $f\neq{\rm const}$ ...
### Solve the functional equation $f(x+1)-f(x)=x*\sin(x)$ [closed]
Solve $f(x+1)-f(x)=x*\sin(x)$