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 ...

learn more… | top users | synonyms

4
votes
2answers
120 views

Prove that $f′(x)=f′(0)f(x)$ derivatives

Let $f:I \to R$ be differrentiable on an open interval $I \subseteq R$ with $$f(a + b) = f(a)f(b) \quad \forall a, b \in R$$ Suppose that $f(0) = 1$ and that $f'(0)$ exists. Show that: $$f'(x) = ...
4
votes
2answers
236 views

solving functional equation $f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$ for all real numbers [duplicate]

The functional equation to be solved is $f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$. Domain: Reals, Codomain: Reals.I found about 4 possible solutions to the equation but ran into a fundamental problem with ...
4
votes
1answer
143 views

Tricky probability problem

I am having trouble with proving the following assertion: $X,Y$ are i.i.d. with mean $0$ and variance $1$. If $X+Y$ and $X-Y$ are independent then $X,Y$ are normally distributed. Should I be ...
10
votes
2answers
253 views

Solving the functional equation $f(xy)=f(f(x)+f(y))$

Find all functions from $f: \mathbb{R} \to \mathbb{R}$ such that for all $x$ and $y$ $$f (xy)=f (f (x)+f (y))$$ I've put $x$ and $y$ as $0$ and $1$. How to proceed after substituting if we don't ...
10
votes
2answers
863 views

find functions f such that $f(f(x))=xf(x)+1$,

let $f:R\longrightarrow R$, and $f$ is continous,and such that $f(f(x))=xf(x)+1$, find all this $f$? follow is my some idea:(but I don't have solution) We have $f(f(0)) = 1$, so there is your $c = ...
7
votes
1answer
165 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
6
votes
2answers
113 views

Functional Equation: When $f(x+y)=f(x)+f(y)-(xy-1)^2$

How does one solve the following functional equation when $f:\mathbb{R}\rightarrow\mathbb{R}$ $$f(x+y)=f(x)+f(y)-(xy-1)^2$$ When I assumed it was a polynomial equation, it can be seen through ...
6
votes
1answer
198 views

An elementary functional equation.

I am finding this functional equation from a past high school mathematics competition rather tricky. Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}^+$ such that: ...
4
votes
3answers
194 views

Solve differential equation $f'(z) = e^{-2} (f(z/e))^2$

I'm curious if there's a simple closed solution to the following DE and, if so, what it is. $$\begin{align} f'(z) &= e^{-2} (f(z/e))^2 \\ f(0) &= 1. \end{align}$$
11
votes
2answers
639 views

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
10
votes
3answers
298 views

Functional Equation (no. of solutions): $f(x+y) + f(x-y) = 2f(x) + 2f(y)$

Find all the functions $f\colon\mathbb{Q} \to \mathbb{Q}$ such that $f(x+y) + f(x-y) = 2f(x) + 2f(y)$, for all rationals $x,y$.
10
votes
4answers
688 views

The functional equation $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$

How to find the all continuous functions $f\colon \mathbb{R}\to\mathbb{R}$ such that $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$
8
votes
1answer
136 views

All functions $g:\mathbb{N}\to\mathbb{Z}$ such that $m+n~|~g(m)+g(n)$

I just thought about the following question: Find all functions $g:\mathbb{N}\to\mathbb{Z}$ such that $m+n~|~g(m)+g(n)$ for all $m,n\in\mathbb{N}$. Clearly every polynomial $g(X)\in\mathbb{Z}[X]$ in ...
8
votes
2answers
396 views

How to prove that $f(f(x))=-x$ implies that $f$ is not continuous? [duplicate]

I am trying to prove that: Given an $f:\mathbb{R} \rightarrow \mathbb{R}$, if $f(f(x))=-x$ then $f$ is not continuous? any help? Thank you!
8
votes
2answers
1k views

Functional Equation $f(x+y)=f(x)+f(y)+f(x)f(y)$

I need to find all the continuous functions from $\mathbb R\rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)+f(x)f(y)$. I know, what I assume to be, the general way to attempt these problems, but I ...
5
votes
1answer
114 views

About $f(x)= f(\frac{1}{x})$

Consider the equation $$f(x)=f\left(\frac{1}{x}\right)$$ Where we want $f$ to be real-meromorphic. Are all solutions $f$ of the form $$f(x) = g\left(\frac{x}{1+x^2}\right)$$ Where $g$ is a ...
3
votes
2answers
135 views

Find $g'(x)$ at $x=0$

The question is: Question. Let $$f(x-y)=f(x)\cdot g(y)-f(y)\cdot g(x)$$ and $$g(x-y)=g(x)\cdot g(y)+f(x)\cdot f(y)$$ for all $x,y \in \mathbb{R} $. If right hand derivative at $x=0$ exists for ...
0
votes
1answer
41 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
9
votes
1answer
94 views

Function preserving exponentiation [duplicate]

I'm wondering what kind of function preserves exponentiation, i.e., what is an $f$ such that $f(a^b)=f(a)^{f(b)}$?
8
votes
1answer
211 views

If $f(2x)=2f(x), \,f'(0)=0$ Then $f(x)=0$

Recently, when I was working on a functional equation, I encountered something like an ordinary differential equation with boundary conditions! Theorem. If the following holds for all $x \in ...
8
votes
4answers
319 views

if the function $f(f(n))+f(n)=2n+2014$,find the $f$

let the function $f:N^{+}\to N^{+}$,and such $$f(f(n))+f(n)=2n+2014$$ Find the $f(n)$ My try: let $n=1$,then we have $$f(f(1))+f(1)=2016$$ let $f(1)=a$,then $$f(a)+a=2016$$ and let ...
7
votes
2answers
143 views

When is it possible to have $f(x+y)=f(x)+f(y)+g(xy)$?

According to the question mentioned here, it seems that there is no function $f(x)$ such that the functional equation $$f(x+y)=f(x)+f(y)-(xy-1)^2$$ can hold. Motivated by this question, I found it ...
7
votes
0answers
132 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
7
votes
4answers
903 views

Find a solution for $f\left(\frac{1}{x}\right)+f(x+1)=x$

Title says all. If $f$ is an analytic function on the real line, and $f\left(\dfrac{1}{x}\right)+f(x+1)=x$, what, if any, is a possible solution for $f(x)$? Additionally, what are any solutions for ...
7
votes
1answer
148 views

Periodic solution of differential equation y′=f(y)

Let $f∈C^∞(ℝ^2,ℝ^2)$ and $∀x∈ℝ^2$ $f(kx)=k^2f(x)$ for $k∈ℝ$ Show that any periodic solution of $y′=f(y)$ is constant. My attempt : Let $\lambda \in \mathbb{R}$. Let $g$ a periodic solution ...
7
votes
1answer
193 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
7
votes
3answers
195 views

Is every function $f : \mathbb N \to \mathbb N$ a composition $f = g\circ g$?

True or wrong: For every function $f: \mathbb N \rightarrow \mathbb N$ there is a function $g: \mathbb N \rightarrow \mathbb N$ with $f=g \circ g$.
7
votes
2answers
235 views

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$ and prove they are indeed all. Is there an easy way to prove this?
7
votes
3answers
308 views

$f(x^2) = 2f(x)$ and $f(x)$ continuous

I ran into a problem recently where I obtained the following constraint on a function. $$f(x^2) = 2f(x) \,\,\,\, \forall x \geq 0$$ and the function $f(x)$ is continuous. Can we conclude that $f(x)$ ...
7
votes
3answers
212 views

Is there a non-constant function $f:\mathbb{R}^2 \to \mathbb{Z}/2\mathbb{Z}$ that sums to 0 on corners of squares?

A problem in the 2009 Putnam asks about functions $f:\mathbb{R}^2 \to \mathbb{R}$ such that whenever $A,B,C,D$ are corners of some square we have $f(A)+f(B)+f(C)+f(D)=0$. Without spoiling the problem ...
5
votes
1answer
172 views

Concerning nonlinear functional equations

There's a problem I've been working on for awhile that involves some hefty functional equations. For example, I may have something along the lines of $$ f(x)f(x) =x+1+f(x+1) $$ I've tried several ...
4
votes
2answers
195 views

Functional Equation : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x).

Problem : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x). My approach : The given equation can be written as $$(x-y)f(x+y) -(x+y)f(x-y) =4xy(x-y)(x+y)$$ $$\Rightarrow ...
3
votes
2answers
896 views

To find an extremal of the functional $\int_0^1 [(y')^2 + 12 xy] dx$

I have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial ...
2
votes
3answers
262 views

$1992$ IMO Functional Equation problem

The problem states: Let $\Bbb R$ denote the set of all real numbers. Find all functions $f : \Bbb R \rightarrow \Bbb R$ such that $$f(x^{2}+f(y))=y+(f(x))^{2} \space \space \space \forall x, y \in ...
2
votes
2answers
245 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
15
votes
1answer
2k views

Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$

Here is the problem: Find all continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy $$f(x+y)+f(x-y)=2[f(x)+f(y)]\;\;\;(1).$$ Here is my attempt: Fix $\delta>0$ and let ...
14
votes
3answers
872 views

very elementary proof of Maxwell's theorem

Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
9
votes
2answers
356 views

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can't see where we ...
8
votes
1answer
98 views

Complicated real to real functional equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x)+1)^2$$ for all $x,y \in \mathbb{R}.$ So far I have proved that $f$ is bijective. How should I continue?
8
votes
2answers
410 views

How to find all polynomials P(x) such that $P(x^2-2)=P(x)^2 -2$?

I am trying the fallowing exercise : Solve $P(X^2 -2)=P(X)^2 -2$ with P a monic polynomial (non-constant) My attempt : Let P satisfying $P(X^2-2) = (P(X))^2-2$ Then $Q(X)=P(X^2-2) = (P(X))^2-2$ ...
8
votes
2answers
193 views

Continuous $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the continuous functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
7
votes
3answers
456 views

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 ...
7
votes
1answer
490 views

How to find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$?

Can someone please show me how to: Find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$? I've tried substitiuting $x=0,1$. Can't seem to figure it out. The square on the RHS is confusing ...
6
votes
2answers
705 views

Real Analysis Proofs: Additive Functions

I'm new here and could really use some help please: Let $f$ be an additive function. So for all $x,y \in \mathbb{R}$, $f(x+y) = f(x)+f(y)$. Prove that if there are $M>0$ and $a>0$ such that ...
6
votes
2answers
574 views

Uniqueness of Tetration

Let $f(0)=1$ and $f(x+1)=2^{f(x)}$ Also let f be infinitely differentiable. Then does f exist and is it unique? If f is merely continuous, then any continuous function such that f(0)=1 f(1)=2 ...
5
votes
2answers
272 views

Cauchy's functional equation for $\mathbb R^n$

Suppose $f(x+y)=f(x)+f(y)$. If $f:\mathbb R\to \mathbb R$ and is measurable, then $f(x)=cx$. This is referred to as Cauchy's functional equation. Suppose $f:\mathbb R^n\to \mathbb R^n$ instead. Does ...
4
votes
6answers
65 views

Not Understanding a specific substitution rule

I was given the question, If $f(3x+5) = x^2-1$, what is $f(2)$? I am trying to understand the reasoning why $3x+5$ is set equal to $2$.
4
votes
1answer
605 views

About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$

Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which satisfies $$f(xy)=f(x)f(y)-f(x+y)+1$$ for all $x,y\in\mathbb Q$. With a supp condition $f(1)=2$. (I didn't notice that.) How to show that ...
3
votes
1answer
57 views

What “natural” functions besides $g(x) = x$ and $g(x) =\dfrac{c^{x}}{c^{x}+c^{1/2}} $ satisfy $g(x)+g(1-x) = 1$?

I'm not quite sure how I should state this question. This is one way: What "natural" functions besides $g(x) = x$ and $g(x) =\dfrac{c^{x}}{c^{x}+c^{1/2}} $ satisfy $g(x)+g(1-x) = 1$? By "natural" I ...
3
votes
1answer
68 views

Functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$

How to find all functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ for $x> 1$?