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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Finding a function which satisfies an equation

During my analysis course, I found this interesting problem: Let $f: A \to A$ a function such that $f(f(x))+3x=4f(x)$ for every $x \in A$, where A is a finite set of positive real numbers. Find the ...
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2answers
47 views

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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2answers
36 views

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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1answer
51 views

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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1answer
39 views

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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1answer
17 views

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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1answer
50 views

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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1answer
62 views

Functional Equation - Rational

Fing all functions $g: R \to R$ such that, $g(x+y) + g(x)g(y) = g(xy) + g(x) + g(y)$ I have shown that $g(x) = 0$ for all $x$ and $g(x) = 2$ for all $x$ are solutions. I have also show that $g(x) = ...
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3answers
49 views

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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1answer
99 views

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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0answers
29 views

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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41 views

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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2answers
227 views

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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30 views

How can I determine the best relationship for 3 variables, given several data points?

What is the best way to determine the relationship for three apparently related variables? The relationship does not appear to be linear, and may follow a combination of non-linear functions. I have ...
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1answer
38 views

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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1answer
19 views

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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5answers
92 views

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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2answers
33 views

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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0answers
42 views

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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24 views

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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1answer
29 views

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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37 views

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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2answers
46 views

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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1answer
46 views

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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21 views

Functional equation with only two solutions?

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 ...
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2answers
62 views

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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1answer
21 views

Functional equation: $f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$

Find an algebraic function $f:\Bbb N\to\Bbb N$ such that $$f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$$ and $$f(1)=1$$ for all $x\in\Bbb N$ I allready know two things: $f(p^k)=p$ where $p$ is prime ...
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1answer
42 views

Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that $f(x)=f(x^y)$

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$f(x)=f(x^y)$$ for all $x,y\in\mathbb{N}$. I'm not intrested in the trivial solution $f(x)=k$, where $k\in\mathbb{N}$.
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22 views

System of linear Volterra integral equation

Consider the Volterra integral equation $$ f(t) = g(t) +\int_0^t K(t,s) f(s) ds $$ where $g(t)$ is continuous on $t\in[0,T]$ and $K(t,s)$ is a weakly singular kernel. It is well known that there ...
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1answer
82 views

Functions satisfying 4 out of 5 inner product properties

Let us consider function $s:K^m \times K^m \mapsto K$ (here $K = \mathbb{R}$ or $K = \mathbb{C}$). If $\forall x, y, z \in K, \forall \lambda \in K$ $s(x + y, z) = s(x, z) + s(y, z)$ $s(\lambda x, ...
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2answers
50 views

Polynomial equation

Is it true that if $P \in \mathbb{Z}[X]$, then there exists $Q \in \mathbb{Z}[X]$ such that $P(x)=Q(x+1)-Q(x)$? Can we generalize to other rings than $\mathbb{Z}$? I came up with this by ...
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0answers
33 views

Find $f$ such that $(f(x)-f(y))\left( f\left(\frac{x+y}{2}\right) -f(\sqrt{xy})\right) =0,\forall x,y \in (0,\infty).$

Find all continuous function $f:(0,\infty)\to\mathbb{R}$ such that $$\displaystyle (f(x)-f(y))\left( f\left(\frac{x+y}{2}\right) -f(\sqrt{xy})\right) =0,\forall x,y \in (0,\infty).$$ My try: Assume $...
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1answer
66 views

if $f(x + y) = f(x)f(y)$ is continuous, then it has to be injective.

Let $f$: $\Bbb R$ $\rightarrow$ $\Bbb R$ be a non-constant function such that $f(a + b) = f(a)f(b)$ for all real numbers $a$ and $b$. Prove that if $f(x + y) = f(x)f(y)$ is continuous, then it has ...
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32 views

Show that $f$ is continuous and hence of the form $f(x)=cx$ [duplicate]

Let $f:\Bbb R\to \Bbb R$ be additive i.e. satisfies $f(x+y)=f(x)+f(y)$ for all $x$, $y$. and Lebesgue measurable. Show that $f$ is continuous and hence of the form $f(x)=cx$. In order to show ...
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28 views

Functional differential equation

I want to solve a functional differential equation of this kind $$ \int d t'' dt''' a(t,t'') b(t'',t''') \frac{\delta u(t'',t';[x])}{\delta x(t''')}+\int d t'' c(t,t'') x(t'') u(t'',t';[x])+u(t,t';[\...
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63 views

Solutions for differential equations of the form: $f = f' \circ f''\circ \ldots \circ f^{(n)}$

Which are the n-times differentiable real functions that fit the condition: $f = f' \circ f'' \circ \ldots \circ f^{(n)}$ ? I think I have came up with a tentative solution for $n=2$, which may ...
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21 views

Solving equation over expectation

I have an expression $(1 + n EX^k)p^{-k}$ which I would like to minimize over $k$. Here $n$ and $k$ are positive numbers, while $X$ and $p$ are in $[0,1]$. Since the expectation converges absolutely, ...
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0answers
20 views

Interval between two solutions of an equation

We are in $\mathbb{R}$ and $x\geq 0$. I have an equation: $(1-\alpha)x^{\frac{1}{1-\alpha}}-W(y)x=g(y)W(y)$ Where $\alpha \in (0,1)$, y is a parameter $0<y<1$, W and g are constants in $x$ and ...
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0answers
30 views

Solving the functional equation F(z+1)=Q(z)F(z)

I want to solve the functional equation $F(z+1)=Q(z)F(z)$. The $F(z)$ is a matrix function. $Q(z)$ is also a matrix function. But its compoents are all rational functions. i.e. $ F(z)=\begin{matrix} ...
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11 views

Pacejka formula - extracting the curve peak

I don't know if many of you knows the Pacejka Magical Formula, but it looks like this: $D\sin(C\arctan(Bx - E(Bx-\arctan(Bx))))$ As you can see, the formula reaches a peak point(in this case at ~0.6)...
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1answer
28 views

Multiplicative even and odd functions?

An even function satisfies $$ f_e(x) = f_e(-x) $$ and a odd function $$ f_o(x) = -f_o(-x) $$ Every function can be split into an even and an odd part $$ f(x) = f_e(x) + f_o(x) = \frac{1}{2}(f(x)+f(-x))...
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22 views

How to complete this?

Let $f$ a function defined on $]0,+\infty[$ and checking : $$\forall x,y>0,\ f(xy)=f(x)+f(y)$$ Assume that $f$ is bounded on $]1-\eta,1+\eta[$($\subset]0,+\infty[$). Let $\alpha =\underset{x\in ...
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0answers
37 views

Functional analysis with KKT conditions

I want to solve an optimization problem min $F(x_{ik}) $ subject to $x \in X$. $F$ here is a function or functional that I wish to determine. I want my optimal ...
5
votes
3answers
73 views

$f(x)$ is an analytic function in $\mathbb{R}$ such that $f(-x)f(x)=1$. What else can we find out about $f(x)$?

Well, I know that there are some easy things we can say immediately: $f(0)= \pm 1$, follows immediately $f(x)=\pm 1$ is the obvious solution, so let's look for other solutions. Moreover, let's ...
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27 views

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 $(f,g)=(\sin{ax},\cos{ax})$...
5
votes
2answers
70 views

Can a conformal map be turned into an isometry?

Let $f: (M, g) \to (M, g)$ be a conformal diffeomorphism of the riemannian manifold $(M, g)$, with $$ g(f(p))(Df(p) \cdot v_1, Df(p) \cdot v_2) = \mu^2(p) g(p)(v_1, v_2), \quad \forall p \in M, \, \...
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0answers
25 views

Solving a functional convolution equation

I have a very weird functional equation that I am trying to solve. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a non-negative Lebesgue integrable function such that $\int f=1$. Let $\tilde{f}(x)=f(-x)$ ...
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0answers
50 views

What does non-degenerate mean?

The context is that f is a real-valued function on a plane such that for every non-degenerate square ABCD in the plane, f(A)+f(B)+f(C)+f(D)=0.... What does this word mean here? I've found that "in ...
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0answers
48 views

Find all functions $f: \Bbb Q \rightarrow \Bbb Q$

Find all functions $f:\Bbb Q \rightarrow\Bbb Q$ that satisfy the conditions: a) $f(f(2016))=0;$ b) $f(x+y)= f(x)+f(y)+f(2016)$ for all $x,y \in \Bbb Q.$ I tried to solve the problem using the ...
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1answer
47 views

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): ...