# Tagged Questions

The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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### Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$

Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$. We have that $f(0) = 0$ and $f(x+1) = f(x)+f(1)+2x$ and thus $f(x+1) - f(x) = f(1)+2x$. Then we see ...
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### Prove that there are no two functions $f$ and $g$ defined over real numbers that satisfy either of the following functional equations

Prove that there are no two functions $f$ and $g$ defined over real numbers that satisfy either of the following functional equations: $f(x)g(y) = x+y$ and $f(x) + g(y) = xy$. I think we should ...
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### If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$

Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Show that if $f$ is continuous, then $f(x) = cx$ for all $x \in \mathbb{R}.$ I find it hard ...
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### Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$

Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$. We know ...
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### Can I prove that $f(x,y)$ can be written as $g(x+y)$ under certain conditions.

I have $f(x,y):R^2\rightarrow R$. I know $f(x,y)=f(y,x)$ and $f(x+d,y)=f(x,y+d)$. Can I prove that I can express $f(x,y)$ as $g(x+y)$. This is where I got: $f(x+d,y)=f(x,y+d)$, I plug in $x=0$ Gives ...
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### Help finding a $C^{1}$ function, with given $C^{1}$ functions, a relation, and some additional assumptions.

I'm down to the following problem (see below) that I just need some insight on (I couldn't find anything close to help online, via other posts here, etc.). My initial thoughts were to use the Inverse ...
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### Functional equation $(n-1)^2 < f(n) f(f(n)) < n^2 +n$.

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$(n-1)^2 < f(n) f(f(n)) < n^2 +n$$ for all $n \in \mathbb{N}$.
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### Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$:$f(f(f(x)+y)+y)=x+y+f(y)$

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$: $f(f(f(x)+y)+y)=x+y+f(y)$ I got the following: (1)$f$ is injective (2) $f(0)=0$ (3)$f(f(f(x)))=x$ But then ...
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### Let $f$ be a function from $Q$ to $Q$ satisfying $f(x+f(y))=f(x).f(y)$. Prove that $f$ is constant

Let $f$ be a function from $Q$ to $Q$ satisfying $f(x+f(y))=f(x).f(y)$. Prove that $f$ is constant I was trying to divide into 3 cases: when f(x) has a root, when f(x)>0 and when f(x)<0 .. But I ...
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### A functional equation for a Dirichlet series

I'm looking for a functional equation for the following Dirichlet series $$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$ where $q$ is a rational number. Any help ?
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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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### Can this Delay Differential Equation (DDE) be solved in closed form?

$$\frac{\mathrm{d}\psi}{\mathrm{d}x}=\frac{(x+\mu)^2-\lambda^2}{\mu}\psi(x+\mu)$$ where $\mu\geq0$, $\lambda\in\mathbb{C}$ are constants and $\psi(x)\to0$ as $|x|\to\infty$.
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### Functional equation in $a,x,y$

Let $f:(0,+\infty)\rightarrow \mathbb{R}$ and $a>0$ such that $f(a)=1$. Prove that, if \begin{align*} f(x)f(y)=f\left(\frac{a}{x}\right)f\left(\frac{a}{y}\right),\quad\forall x,y>0 \end{align*} ...
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### Functional equations with nowhere differentiable solutions

As an example, the functional equation $f(x+y)=f(x)f(y)$, by declaring that $f$ is continuous and differentiable, we can arrive at the unique solution $f(x)=a^x$, by first showing that $f'(x)=f(0)f(x)$...
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### Prove that there is a differentiable function $f: \mathbb{R} \to \mathbb{R}$ satisfying $[f(x)]^5+f(x)+x = 0$

Prove that there is a differentiable function $f: \mathbb{R} \to \mathbb{R}$ satisfying $$[f(x)]^5+f(x)+x = 0$$ for all $x \in \mathbb{R}$. Find $f'(x)$. Seeing how this is a functional equation, I ...
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### Question about solutions of nonlinear functional equation

One of the basic nonlinear functional equations is the following one: $$(f(x))^{2}=xf(2x),\ \ \ x>0.$$ I found out that functions $f(x)=2^{1-x}x\exp(cx)$ form the family of solutions of this ...
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### Does mathematical modeling assist in learning how to derive equations?

I am hoping someone can provide a starting point to the question: Would mathematical modeling be a good place to start learning how to derive equations and functions from a set of data? For example, ...
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### What can we say about functions satisfying $f(a + b) = f(a)f(b)$ for all $a,b\in \mathbb{R}$? [duplicate]

Possible Duplicate: Is there a name for such kind of function? I am investigating functions satisfying the exponentiation identity $f(a + b) = f(a)f(b)$ for all $a,b\in \Bbb R$. This is ...
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### Find all functions $f: \mathbb N \rightarrow \mathbb N$ such that $f(n!)=f(n)!$

Find all functions $f: \mathbb N \rightarrow \mathbb N$ (where $\mathbb N$ is the set of positive integers) such that $f(n!)=f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m)-f(n)$...
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### Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
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### Different type of function equation

I've spent three days revising functions, although it wasn't enough at all because I think my sources weren't good enough to allow me to become a master in the field of math. By the way, there are two ...
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### what is the relation between $f(x+1)$ and $f(x)$?

I searched so much over math sites and google but I didn't find helpful hints and required knowledge or the specific name of this topic in function. I stuck in relation and operations on $f(x)$ which ...
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### Functions that satisfy the identity $f\left(\frac{x}{t}\right) f\left(-\frac{y}{r}\right)=f\left(\frac{x-y}{t-r}\right)$

I am looking for function(s) which satisfy the following property: $$f\left(\frac{x}{t}\right) f\left(-\frac{y}{r}\right)=f\left(\frac{x-y}{t-r}\right)$$ I am not sure if there is any function ...
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### Resolution of equation such that $f(…f(x)…)=x$

I am wondering if it exits a way to find "easely" the solutions of an equation about a function $f$ such that $$f^{(n)}(x)=x$$ where $f^{(n)}$ is the n-th composition of $f$ itself. Obviously the ...
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### Integration involving functional equations [closed]

Let $f$ be a function satisfying $f(x+y) = f(x) f(y)$ with $f(0) = 1$ and $g$ be a function that satisfies $f(x) + g(x) = x^2$. Then the value of the integral $\int \limits _0 ^1 f(x) g(x) \textrm d x$...
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### Question regarding the Cauchy functional equation

Is it true that, if a real function $f$ satisfies $f(x+y) = f(x) + f(y)$ and vanishes at some $k \neq 0$, then $f(x) = 0$? Over the rationals(or, allowing certain conditions like continuity or ...
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### Functional division $\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$

As the title suggests, the problem here is: Find all functions $f:\mathbb{Z}\to\mathbb{N}$ such that, for every $x,y\in\mathbb{Z}$, we have $$\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$$ I ...
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### Notation without cases? $f(x)=\begin{cases}p,&\text{if$x=p^k$}\\1,&\text{otherwise}\end{cases}$

Is there any other way to write the function $f:\Bbb N\to\Bbb N$ such that $$f(x)=\begin{cases}p,&\text{if x=p^k}\\1,&\text{otherwise}\end{cases}$$ when $p$ is prime and $k\in\Bbb N$?
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### The Functional Equation $f(mn)=f(m)f(n)$ where $f:\mathbb{N}\rightarrow \mathbb{R}$, $f(2)=2$, and $f(m) > fn)$ if $m>n$.

The following is exercise 3.3 from Terence Tao's "Solving Mathematical Problems." Emphasis added. Suppose $f$ is a function on the positive integers which takes real values with the following ...
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### Functional equation with sqrt solutions [closed]

Let $f:(0,\infty)\to(0,\infty)$ so that for all $x,y\in(0,\infty)$ we have $$f\left(\frac {x} {f(y)}\right)=\frac {x} {f(x\sqrt y)}.$$ Find function $f$.
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### Functional equation that models trigonometric identities

Find, with proof, all continuous functions $f,g : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x)^2 + g(x)^2 = 1$ and $2f(x)g(x)=f(2x)$. I am aware that the solution pair $(f,g)=(\sin{ax},\cos{ax})$...
### How to solve $2f(x)+xf(\frac{1}{x})-2f(|\sqrt2 \sin(\pi(x+\frac{1}{4}))|) = 4 \cos^2 \frac{\pi x}{2}+x\cos\frac{\pi}{x}$
I saw a question today. $$2f(x)+xf(\frac{1}{x})-2f(|\sqrt2 \sin(\pi(x+\frac{1}{4}))|) = 4 \cos^2 \frac{\pi x}{2}+x\cos\frac{\pi}{x}$$ It had options like this (one or more than one may be correct): ...
Recently I started studying functional equations. Now I'm trying to find all solutions to the following functional equation: $$(f(2x))^3=f(4x)((f(x))^2+xf(x)).$$ Unfortunately, I was able to show only ...