# 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).

2,509 questions
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### A function describes $g(x + y) = g(x)g(y)$ for all $x, y$. If $g(4) = + 3,$ find the value of $g(–8)$? [closed]

I tried solving the question, but I always ended up getting my answer wrong. I'm also not sure if the given options are correct. Here are the options that were given: A. 1/3 B. 1/9 C. 9 D. 6
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### Solving $\sum\limits_{k=1}^n e(x-x_k) = h(x)$ for $e(x)$, where $x_k$ and $h(x)$ are given

I would like to find the function $e(x)$ which solves $\sum\limits_{k=1}^n e(x-x_k) = h(x)$, where $x_k$ and $h(x)$ are given. There are no restrictions on any of the $x_k$ or $h(x)$ except that $h(x)$...
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### Solution of advanced functional differential equation

Statement Consider an advanced functional differential equation $$Lf(x) = f(2x+\pi)+f(2x-\pi),\quad L\equiv\frac{d^2}{dx^2}+1. \tag{1}$$ Let's construct a solution of Eq. $(1)$ with finite ...
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### Find all continuous functions in $0$ that $2f(2x) = f(x) + x$

I need to find all functions that they are continuous in zero and $$2f(2x) = f(x) + x$$ About I know that there are many examples and that forum but I don't understand one thing in it and I ...
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### Functional equation with $f(2x)$

Any other solutions(advice) are welcome. For any $x>0, \;\;\; 2f(\frac{1}{x}+1)+f(2x)=1$ Find all possible f(x). I wish you luck on a good thing in $2019$.
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### $t(n) = t( x_1 x_2 x_3 …) = t(x_1) + t(x_2) + t(x_3) + … + t( x_1 + x_2 + x_3 + … )$

Let $n > 1$ be an integer. Consider The prime factorization $$n = x_1 x_2 x_3 ...$$ Now define $$t(n) = t( x_1 x_2 x_3 ...) = t(x_1) + t(x_2) + t(x_3) + ... + t( x_1 + x_2 + x_3 + ... )$$...
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### Fixed point of unusual integral equation

I am a little rusty in this area so please forgive the slowness. I am trying to prove or disprove the existence of fixed points for the following integral equation. Throughout I am interested in the ...
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### Find all continuous function $f: \mathbb R \rightarrow \mathbb R$

Find all continuous function $f: \mathbb R \rightarrow \mathbb R$ for which $f(3)=5$ and for every $x,y \in \mathbb R$ it is truth that $f(x+y)=2+f(x)+f(y)$. I tried to find some dependence before $x$...
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### $f(x)f(\frac{1}{x})=1\hspace{1 em}\forall x\in\mathbb{R}$ [duplicate]

Find all functions $f:\mathbb{R^*}\to\mathbb{R^*}$ that satisfy $$f(x)f(\frac{1}{x})=1\hspace{1 em}\forall x\in\mathbb{R^*}$$ I only found that every function $f(x)=x^n, n\in\mathbb{N^*}$ is a ...
Let $f$ and $g$ two rugular functions. My question is the following: Under what condition can we say that for given $g$, there exists $f$ such that we have: $$\int\limits_0^1 {f(x - s,s)ds = g(x)}$$ ...