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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 (...

7
votes
2answers
69 views

Find function $f(x)$ that satisfying differential relation

Suppose the functions $F(x)$ and $G(x)$ satisfying $$F(x)=f(x)-\frac{1}{f(x)}$$ $$G(x)=f(x)+\frac{1}{f(x)}$$ such that $F'(x)=(G\circ G)(x)$, with initial condition $f(\frac{\pi}{4})=1$ is given....
-2
votes
1answer
32 views

Functional Eq. $f(x)=f(3x)*\frac{1}{2}$ [on hold]

Functional Eq. $f(x)=\frac{1}{2}f(3x)$ I have no idea how to solve this. But answer is $f(x)=c_{1}2^{log_{3}x}$
0
votes
1answer
35 views

Find all the bijective functions $f:[0,1]\to[0,1]$ such that $x=\frac{1}{2}\big(f(x)+f^{-1}(x)\big)$ for all $x\in[0,1]$.

Find all bijective functions $ f : [0,1] \to [0,1]$ that satisfy the equation $$x=\frac{1}{2} \big(f(x) +f^{-1} (x)\big)\,\forall x \in[0,1]\,.$$ I honestly don't know how to approach this. ...
13
votes
1answer
154 views

$f(a)-f(b)$ is rational iff $f(a-b) $ is rational

Prove that the continuous function $f:\mathbb{R} \to \mathbb{R}$ satisfying $f\left(x\right)-f\left(y\right) \in\mathbb{Q} \iff f\left(x-y\right) \in \mathbb{Q}$ is of the form $ f\left(x\right)=ax+...
2
votes
2answers
46 views

If $f:\mathbb{R}\to\mathbb{R}$ is infinitely-differentiable, and $f(x+y)-f(y-x)=2xf^\prime(y)$, then it is a polynomial of degree less than $2$

$S$ is set of family of infinite differentiable function from $\mathbb R \to \mathbb R$ with $\forall x,y\in R$ $$f(x+y)-f(y-x)=2xf^\prime(y)$$ then I have to prove that $S$ only contain all ...
0
votes
3answers
34 views

$x_i \frac{\partial f(x)}{\partial x_i} = f(x)g_i(x)$ for all $i$

What can we say about the function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ if for all $i=1,...,k$ and all $x \in \mathbb{R}^k$ we have $x_i\frac{\partial f(x)}{\partial x_i} = f(x)g_i(x)$ Is there ...
1
vote
0answers
57 views

¿How to solve this functional equation: $xf(y)+yf(x)=(x+y)f(x)f(y)$? Need some help! [duplicate]

So I've been reading about functional equations and how to solve them. I found a pretty interesting problem (for me) but I think I need some help, some hint. I've never worked with this kind of ...
6
votes
1answer
68 views

Solving a functional equation arising from a probability problem.

I am trying to find solutions to the following functional equation: $$(g(kx))^2=g(x).$$ Here, $x$ is in $\mathbb{R}$ and $k$ is a constant. In particular, I'm looking for solutions for $k=2^{-1/4}$. ...
2
votes
1answer
61 views

Is there a function that satisfies $f(\log \frac{p}{1-p}) = p(1-p)$?

Let $p\in (0,1)$ I am wondering there exists a function $f:\mathbb{R}\to[0,{1\over 2}]$ that satisfies $$ f\left(\log \frac{p}{1-p}\right) = p(1-p) $$ I've tried messing around with exponentials ...
1
vote
1answer
33 views

Problem in solving a functional equation

I have the following functional equation with me: $f(x-y)=f(x)f(y)+g(x)g(y)$ $f(x)=\cos x $ and $g(x)=\sin x$ is an obvious solution but I have some trouble proving it. I have obtained the following ...
13
votes
1answer
243 views

Polynomials $P(x)\in k[x]$ satisfying condition $P(x^2)=P(-x)P(x)$

This question is inspired by this thread which is on hold at the moment. Fix a field $k$. Let $P(x)\in k[x]$ be such that $$(1)\ \ \ \ \ P(x^2)=P(-x)P(x).$$ Let $T(k,d)\subseteq k[x]$ denote the set ...
6
votes
2answers
152 views

Find function $f$ given $f(x+1) - f(x) = \frac{1}{x^2}$

I need to find the expression of function $f$, all we know about $f$ is: $\begin{cases} \forall x>0, f(x+1)-f(x) = \frac{1}{x^2} \\ f \text{ is continuous on } ]0, +\infty[ \text{ and } \lim\...
5
votes
2answers
66 views

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$ I don't know where to start as in is there a ...
5
votes
3answers
103 views

Functions satisfying $f(a+b) =\frac{f(a)+f(b)}{f(a)*f(b)}$

i was looking for a function that satisfies $f(a+b) =\frac{f(a)+f(b)}{f(a)*f(b)}$ for all $a,b \in \Bbb{N}$. I have never seen such a problem before and i would like some kind of help to get me ...
1
vote
1answer
19 views

$G(x,y)G(y,z)$ independant of $y$ $\implies$ Most general function form $G(x,y) = rH(x)/H(y)$ , $r$ a const.

I am reading Probability Theory, The Logic of Science by E.T. Jaynes; and encountered the following statement in his derivation of the product/chain rule (of probability) from basic principles: $$(1)\...
0
votes
2answers
96 views

Which continuous function $f$ satisfies $f(a+b)={f(a)+f(b)\over 1−f(a)f(b)}$? [closed]

If $f:(-\pi/2,\pi/2) \to\mathbb{R} $ and $$f(a+b)=\frac{f(a)+f(b)}{1-f(a)f(b)}$$ and $f$ is continuous at $x=0$, then show that $f$ is continuous on its domain. Where to start?
0
votes
0answers
35 views

How is it possible to minimize a definite integral, if there are three parameters that can be varied in the integrand?

I have been attempting to minimize the following integral: $$ T=\int_{a}^{b}\sqrt{\frac{1+y'^2}{2gy}}dx, $$ knowing that $y$ is of the form: $$ y = a_0 +a_1 x+a_2x^2 $$ and that $y(0)=2$ and $y(\pi)...
0
votes
0answers
50 views

Solving $f(x) = f(g(x))\,h(x)$ for $f$

How do I go about solving equations of the form $f(x) = f(g(x))\,h(x)$ for the function f, given functions $g$ and $h$? Is there maybe some integral transform that I could use? When $h(x)$ is ...
0
votes
0answers
41 views

An inequality involving the Dirichlet eta function

I would be interested in proving the following inequality involving Dirichlet's eta function $\eta(s)$ at different values which, after some numerical investigations, I am sure is true $|\chi(1-s)\...
0
votes
1answer
71 views

$f(y+zf(x))=f(y)+xf(z) $ with $x,y,z\in \mathbb{R}$

Find all $f\colon\mathbb{R} \rightarrow \mathbb{R}$ such that $f(y+zf(x))=f(y)+xf(z)$ with $x,y,z\in \mathbb{R}$ When I study functional equation, I have some difficult in solving above equation. ...
1
vote
2answers
119 views

Is there a continuous $f$ satisfying $f(f(x))=-x^3+x$?

Is there a continuous function $f$ defined on real number satisfying: $$f(f(x))=-x^3+x.$$ I'm shame to say it's my homework and I've spend several hours on it. Also, I've tried to construct a ...
1
vote
0answers
10 views

First Variation of CDF inside an Indicator Function

I would like to minimize the functional $\mathcal{F}(\mu) = \int x I(F_\mu(x)\leq\tau) d\mu(x)$. However, I'm don't understand how to find the first variation of the term $I(F_\mu(x)\leq\tau) = I(\mu((...
2
votes
1answer
52 views

Find a strictly decreasing $f:(0,1) \to [0, \infty]$ such that $f(0^{+})= \infty$, $f(1)=0$ and $f(u^2)=2 f(u)$.

Assume that $f$ is a function from $(0,1]$ to $[0, \infty]$ that is strictly decreasing and satisfies $f(1)=0$, $f(0^{+})= \infty$ and $f(u^{2})=2 f(u)$. I do not know if it helps, but let us assume ...
0
votes
0answers
15 views

Pexider's (/ Cauchy's) functional equation over a bounded domain

I am looking at Pexider's equation $f(x+y)=g(x)+h(y)$, where $f,g,h$ are continuous functions but are defined over bounded domains. Specifically, $f,g,h$ each is defined on a real interval (of length ...
-1
votes
0answers
27 views

finding the properties of function or function itself

find all functions that satisfy $f(x+y)+f(x-y)=f(x)f(y)$ $f(x+y)=f(x)+f(y)-2f(xy)$ 1,2 are independant. Is 1 periodic function?
3
votes
2answers
422 views

Prove that, the function $f$ is injective: $f \big(f(x)f(y)\big) + f(x +y) = f(xy).$

I need to learn, by which method, I can prove that the function $f$ is injective. I would like to ask you to explain this problem using more detailed, more understandable, clearer and simpler ...
0
votes
1answer
36 views

Functional Identity

If we know that $$ f'(k/x)f(x) = x\tag{ * } $$ Then what can we say about $$f(k/x)f'(x) ?$$ Originally I tried substituting $x=k/x$ into (*), to give $$f'(x)f(k/x) = k/x$$ But is this valid? I'm ...
3
votes
0answers
71 views

Asymptotics for $ f(n) = f(n - 1) + f( n - g(n) ) $?

Define $g(x)$ as : If $f(m) =< x < f(m+1)$ for a positive integer $m$ then $g(x) = m$. Now we define $f(n)$ for strict positive integer $n$. $$f(1) = 1 $$ $$f(2) = 3 $$ $$f(3) = 7 $$ For $n &...
2
votes
3answers
60 views

Probability density function invariant under $x\mapsto 1/x$

Let $f(x)$ be a probability density function (pdf) over $(0,\infty)$, i.e., $f(x)\geq 0$ and $\int_0^\infty dx\ f(x)=1$. Is it possible to characterise the set of such pdfs for which the following ...
13
votes
1answer
444 views

Determining if a differential equation has unique solution

Our teacher asked a seemingly simple question: What is the value of $f(2)~$ if $~f( f(x) ) = 16x-15~$? I found a solution assuming $f(x)$ in the form of $ax+b$, but how do I show that $f(x)$ must ...
3
votes
0answers
39 views

Functional equation for the lambert w function?

I've been long with this question or trying to find something similar: Is there a functional equation of a reflection formula for the Lambert W function? The Lambert W function is the inverse ...
7
votes
2answers
244 views

Prove that $f$ is additive if $f(x)f(x-y)+f(y)f(x+y)= f(x)^2+f(y)^2$

Say $f:\mathbb{R}\to \mathbb{R}$ is non-constant such that $$f(x)f(x-y)+f(y)f(x+y)= f(x)^2+f(y)^2$$ Prove that $f(x+y)=f(x)+f(y)$. If we put $a=f(0)$ and $y=0$ we get $$f(x)^2+af(x)= f(x)^2+a^2 \...
5
votes
4answers
177 views

Find all functions such that $f(x^2+y^2f(x))=xf(y)^2-f(x)^2$

I am dealing with the test of the OBM (Brasilian Math Olympiad), University level, 2016, phase 2. I hope someone can help me to discuss this test. Thanks for any help. The question 2 says: Find ...
3
votes
0answers
39 views

Integral equation in polar coodinate system

I need an inversion formula with the form $f(r)=\cdots$, from this integral relation: $$g(r)=\frac{1}{2\pi}\int_0^{2\pi}d\theta\,f\left(\sqrt{r^2+r_0^2-2rr_0\cos\theta}\right)$$ where $r_0\geq0$ is a ...
1
vote
1answer
34 views

Relation between CDF and PDF

I want to prove there is no random variable X that satisfies the following relation: $$f_X(x) = \bar{F}_X(x)\bar{F}_X(-x)F_X(x)$$ where $f_X$, $\bar{F}_X$ and $F_X$ are the PDF, Complementary CDF and ...
0
votes
3answers
69 views

solving $f(x)=\frac x{1-x^2}+2(1+x)f(x^2)$ without power series expansion

My question concerns the title equation $f(x)=\frac x{1-x^2}+2(1+x)f(x^2)$, which arose through the use of generating functions for a simple recurrence. Assuming $f: R\rightarrow R$ is analytic, the ...
1
vote
1answer
16 views

Help understanding an injectivity proving technique in functional equations

In need help understanding this (It's from Evan Chen's Introduction to Functional equations) When trying to obtain injective or surjective, watch for "isolated" variables or parts of the equation. ...
1
vote
1answer
29 views

Functional equation - Cyclic Substitutions

Please help solve below functional equation $f: \mathbb R \rightarrow \mathbb R$ $f(-x) = -f(x) , f(x+1) = f(x) + 1, f\left(\frac 1x\right) = \frac{f(x)}{x^2}$ for all $x \in \mathbb R$ and $x \ne 0.$...
0
votes
1answer
31 views

Determine all continuous real function which satisfies the following [duplicate]

We are required to determine all continuous real valued functions $f$ such that $$f(f(x))=-x$$ I’ve determined that if such a function exists, it must be bijective. But I don’t know if such a ...
6
votes
1answer
79 views

Existence of two functions $f$ and $g$ for which $f\circ g (x)=x^2 , g\circ f (x)=x^3$

Do there exist two functions $f$ and $g$ from the reals to itself satisfying $f\circ g (x)=x^2 , g\circ f (x)=x^3$ for any $x\in\mathbb{R}$? From the given equations I could get the following ...
0
votes
2answers
95 views

Does this a function $x+y \rightarrow x^2+y^2$ exist?

I don't know if this exists, but it would make my algebra easier if it did instead of having to use complicated radicals to solve an equation. If $x \in \mathbb{R}$ and $y \in \mathbb{R}$ and $h(x,y)=...
2
votes
0answers
34 views

When is adding a random variable to itself equivalent to multiplying the random variable by two?

Let $X$ and $Y$ be independent random variables with the same distribution over the reals. When does $X+Y\sim2X$ hold? Edit: By "$\sim$" I intend to express that the left and right hand sides have ...
0
votes
1answer
25 views

Functional derivative normalization sensitive to normalization of test function?

Background: I am a physicist with decent background in mathematics. Reading the article on functional derivatives on wikipedia gives: The functional derivative $\frac{\delta F}{\delta \rho(x)}$ of ...
7
votes
3answers
550 views

IMO 2017: Determine all functions $f: \mathbb{R} \to\mathbb{R}$ such that, for all real numbers $x$ and $y$, $f(f(x)f(y)) + f(x +y) = f(xy)$.

EDİT: I think I've repair the error in the solution. I want to know if I'm fixing it properly. I'm just a student, not a mathematician. Please focus on the "backbone" of my writing. I want to prove ...
0
votes
1answer
49 views

Solving functional equations using Partial Derivatives(High School Level)

$$f(x)+f(x+2y)+3xy=2f(2y-x)+2y^2,~x,y\in\mathbb{R}$$ I'm given a functional equation $f(x,y)$. I've been told by my teachers to solve such equations by first partially differenciating the equation ...
3
votes
1answer
36 views

Mean of two i.i.d. random variables follows the same distribution, Cauchy distribution?

It is well known that if $X$ and $Y$ follow i.i.d. Cauchy distribution of scale $\gamma$, say $$ p_{\gamma} (x) = \frac{1}{ \pi \gamma ( 1 + x^2 / \gamma^2 ) }, $$ then their arithmetic mean $ ( X + Y ...
8
votes
1answer
122 views

Solving $f(x) = f(x/2) + f(x/3) + x$

How would one proceed to proving that the solution to the functional equation $$f(x) = f(x/2) + f(x/3) + x$$ is $f(x)=6x$ which is also unique? To clarify, I am not aware of neither the proof of the ...
3
votes
1answer
55 views

Solutions to $-t\ f'(t)+(1+r t)\ f(t)=f(t/2)$

Any thoughts on how to tackle this ODE $-t\ f'(t)+(1+r t)\ f(t)=f(t/2)$, $t\in[0,t_1]$, with boundary conditions $f(0)=0$ and $f(t_1)=s$, with $r,s$ and $t_1$ fixed constant.
0
votes
1answer
37 views

Solving functional equation by comparising sides

I have a brain lag. I'm overthinking probably. I have the following equation: $$ f^{-1}(f(x)+f(y))=(g\circ h)^{-1}((g\circ h)(x)+(g\circ h)(y)), $$ where, say, $f,g,h$ are automorphisms on $\mathbb{R}$...
1
vote
1answer
25 views

Proof that solutions to Cauchy F.E. over $\mathbb{Q}$ are linear

I would like to prove that the solutions to Cauchy's functional equation over $\mathbb{Q}$ are linear, that is, all solutions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ have the property $f(x)=cx$ for some $...