# 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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### What can be $f$ so that $f(f(x)) = -x$? [duplicate]

What can be $f$ so that $f^2(x) = -x$ for all $x\in R$? I know that if $f^2(x) = -x$ then $f(x)$ is injective and $f$ can not be continuous. But I can not find an example of discontinuous function ...
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### Prove that $f(x) = x$ is the unique solution to the following functional equation [closed]

Let $f : [0 , \infty) \to \mathbb{R},$ continuous at $x_0 = 0$ satisfying $$f(3x) - 2x = f(x)$$ Prove that $f(x) = x$.
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### The bivariate function from its marginals

Suppose I have a bivariate function, $f(x,y)$. Fixing $x$, we define $f_x(y)=f(x,y)$. Fixing $y$, we define $f_y(x)=f(x,y)$. Now if we know the form of $f_x(.)$ for all $x$, and the form of $f_y(.)$ ...
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### Searching a formula for scaling/mapping a variable based on 3 known values

I am sending an specc'ed integer (X) between -2048, and 2048 to a synthesizer to control its tuning. When X is 0 = Tuning on Synth is 440 (default) When X is 2048 = Tuning on Synth is 546.42 When X ...
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### How can one show that $f(n)>n$

Given a function $f\colon \Bbb N^\ast\to \Bbb N^\ast$ such that we have $f(f(n))=f(n+1)-f(n)~\forall~n\in\Bbb N^\ast$. Show that: If $f(n+1)-f(n)>1$ then $f(n)>n$. Since $f(n+1)>f(n)+1$ I ...
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### How to solve the functional equation $f(x^2+xf(y))= xf(y)$ [closed]

Hello please how to find all the functions $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+xf(y))= xf(y)$$ I see that $f(0)=0$ but how to do after
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### For the given function $f(2)=5$, find $f(3)$

Given that $f(x)$ is differentiable function of $x$ and that $f(x)\cdot f(y) = f(x) + f(y) + f(xy) - 2$ and that $f(2) = 5$. Then find value of $f(3)$. By putting $x=2$ and $y=1$ , I got $f(1)=2$ ...
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### Prove that $\boldsymbol{\ f}$ is a constant function

if :$$f:[0,1]→\mathbb{R}$$ and for all $x\in[0,1]$ $$f(x)=f\!\left(x^2\right)$$ and for all $b\in[0,1]$ $$\lim_{ x \to b }f(x)=f(b)$$ prove : for some $c\in\mathbb{R}$, for all $x\in[0,1]$, $f(x)=c$...
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### Number of functions in $g(g(x))=g(x)$

A function is defined as $g:\{1,2,3,4\}\rightarrow \{1,2,3,4\}$ such that $g(g(x))=g(x)\forall x\{1,2,3,4\}.$ Then number of such functions are Try: from $g(g(x))=g(x)$ implies $g(x)=x$Identity ...
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### Find all functions $f:\Bbb R \rightarrow \Bbb R$ such that for every $x,y \in \Bbb R$: i) $f(x)\leq x$ ii) $f(x+y)\leq f(x)+f(y)$ [closed]

Find all functions $f:\Bbb R \rightarrow \Bbb R$ such that for every $x,y \in \Bbb R$: i) $f(x)\leq x$ ii) $f(x+y)\leq f(x)+f(y)$ In functional equations, I have trouble when inequality has a ...
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### Integral equation that's cant solve… Need a hand [closed]

Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations. $$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi (y)dy$$...
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### Function involving real values

Let $f$ be a function satisfying $f(x/2+y/2)=(f(x)+f(y))/2$, for all real $x$ and $y$. If $f'(0)$ exists and equal to $-1$ , then $f(2)$ equals: ? I have tried this question by putting various values ...
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### Functional differential equation (FDE): $f'(x)=f(x+e)$

I would like to solve the functional differential equation $$f'(x)=f(x-e)$$ I've solved functional equations before, and I've solved differential equations, but I've never solved a functional ...
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### I am trying to find the function f that satisfies $\cos x=f '(x)+f(-x)$

$\cos x=f '(x)+f(-x)$ I manage to solve $f '(x)+f(x)= \cos x$, starting first by solving $y'=-y$ using $y=\exp(ax)$. But here I get stuck because of $f(-x)$.
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### Nontrivial entire $f(z)$ never equal to $0$ [closed]

I'm looking for nonconstant entire functions $f(z)$ such that $f(z)\neq 0$ for any $z$. More specifically I'm looking for nontrivial cases. So $\exp(z),\exp(z^2),...$ is not what I am looking for. ...
### Find all the $f:\mathbb{R}\rightarrow\mathbb{R}$,satisfying
$f(f^3(x)+y^3)=x^2+f^3(y)$ where $f^3(x)$ stands for $[f(x)]^3$. I really don't know where to start off$\cdots$
The Help I need I am preparing for a competitive math exam. I need to learn methods of solving questions like this If f$\left(\frac{x+y}{3}\right)= \frac{2+f(x)+f(y)}{3}$ for all real $x$ and $y$ ...