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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2
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7answers
134 views

$f\left(x + \frac1x\right)= x^3+x^{-3},$ find $f(x)$

$$f\left(x + \frac1x\right)= x^3+x^{-3},$$ find $f(x)$. What i do know at this state is that.. express x as a function of y : $y= x + 1/x$ $x^2−xy+1=0$ Quad formula: $x= (y ± \sqrt {y^2-4}) / 2$ ...
3
votes
1answer
144 views

How to find the function $f$ that satisfies $f(x, y) = f(x^{-1}, y^{-1})^{-1}$ and $f(x, y)$ is $\approx$ $average(x, y)$?

Fist of all, I'm a programmer, not a mathematician, and I'm sorry for my non native English. And I'm sorry if the question is not appropriate, it is my first time here. Or if the question has no ...
1
vote
3answers
73 views

Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$

Suppose that $f: \mathbb R \to\mathbb R$ satisfies $f(x + y) = f(x) + f(y)$ for each real $x,y$. Prove $f$ is continuous at $0$ if and only if $f$ is continuous on $\mathbb R$. Proof: ...
1
vote
0answers
57 views

How to demonstrate a particular functional equation solution

In order to find a prior probability distribution I have to solve the following functional equation: $$af\left(\frac{a\theta}{1-\theta-a\theta}\right)=(1-\theta+a\theta)^2f(\theta)$$ the solution of ...
3
votes
1answer
70 views

Failing to reproduce specific Functional derivative

I'm failing to reproduce an (indirect) result in a paper, namely $${δF[g]\overδg(x,y,z)}={r^4\over\ell^5} $$ where $F[g]=\iiint \frac{2dxdydz}{\ell g(x,y,z)} $ and $g(x,y,z)=-{\ell^2 \over r^2} $. ...
3
votes
1answer
49 views

Are there algorithms for solving simple functional equations?

So somebody posted yesterday asking a question for continuous solutions $f$ satisfying $f(x+y) = f(x)f(y)f(xy)$. Continuity could be used for a simpler proof but then somebody posted a solution ...
2
votes
0answers
24 views

Integers and funtional equation [duplicate]

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that ...
6
votes
1answer
98 views

Functional Equation $f(mn)=f(m)f(n)$.

If $f: \mathbb N \mapsto \mathbb N$ is one-to-one and $f(mn) = f(m)f(n)$, what is the smallest possible value of $f(999)$? Easily $f(1)=1$, and I think $f(n)=n$ must be the only map, but not able to ...
1
vote
1answer
32 views

Differentiability Problem

Supposing we are given relation that $$f(xy + 1)= f(x).f(y) - f(y) - x +2$$ and also given that $$f(0)=1$$ for a differentiable function then is function one-one onto? I partially differentiated ...
1
vote
2answers
152 views

Functional equation involving gamma function

Recently, I found the following functional equation: $$ n^{nx-1}\cdot\prod_{k=0}^{n-1}{\Gamma{\left(x+\frac{k}{n}\right)}}=\Gamma{(nx)}\cdot\prod_{k=1}^{n-1}{\Gamma{\left(\frac{k}{n}\right)}} $$ Now ...
1
vote
1answer
29 views

Proof involving functional equation

I'm trying to prove that if $$f(x+n)=f(x)f(n)$$ for all $x\in \Bbb R$ and $n \in \Bbb N$, then it also holds for $x,n \in \Bbb R$. One "argument" I came up with was regarding the symmetry. There's no ...
10
votes
2answers
233 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 ...
0
votes
2answers
105 views

Approximating the Digamma fucntion near 1

Peace be upon you, I had the following system of equations to be solved \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ \psi(\beta)-\psi(\alpha+\beta)=c_2 \end{cases} \end{align*} ...
3
votes
2answers
102 views

Additive functional equation

Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$ f(x+y) = f(x) + f(y)$$ and $$ f(f(x)) = x$$ for all $x, y \in \mathbb{R}$ This is one problem involving additive functional ...
2
votes
2answers
92 views

Functional equation - Understading an easy step in my solution.

I am trying to solve the equation and find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that: $f(m+f(n))=f(f(m))+f(n)$ for all $n, m \in \mathbb{N_{0}} $. A reasonable approach to begin with ...
1
vote
2answers
81 views

Find all functions $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, f^n(x)=-x$

I got this problem: Prove that the only function $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, \forall x\in\Bbb{R}, f^n(x)=-x$ where $f^n =f\circ ...
1
vote
1answer
53 views

Functional equation- solving techniques

I'm basically a total novice with functional equations and have some questions regarding the solving technuiqes of them. Although, i'm adware of the lack of general solving methods, I have noticed ...
2
votes
1answer
43 views

How equal are two given numbers

I have two numbers x & y for N different readings and wish to find how close they are from each other and would like to rank the reading in order of they equalness. If I were to have the ...
3
votes
1answer
55 views

Find examples of continuous functions $f:[0,1]\to[0,1]$ that satisfy $\forall x\in[0,1], f(f(x))=f(x)$ other than $f(x)=x$

I got this problem: Find examples of continuous functions $f:[0,1]\to[0,1]$ that satisfy $\forall x\in[0,1], f(f(x))=f(x)$ other than $f(x)=x$. I proved that $f([0,1])=\{x\in[0,1]|f(x)=x\}$, But I ...
0
votes
0answers
24 views

Solving functional equation $b(x)=\int b(xy)f(y)dy$

I want to prove that given a real-valued smooth function $f$, the set of functions $b$ solving $b(x)=\int_0^{\infty} b(xy)f(y)dy$ is given by linear combinations of $x^{\sigma}$ where $\sigma$ is a ...
1
vote
4answers
129 views

Example of a non-trivial function such that $f(2x)=f(x)$

Could you give an example of a non-constant function $f$ such that $$ f(x) = f(2x). $$ The one that I can think of is the trivial one, namely $\chi_{\mathbb{Q}}$, the characteristic function on the ...
4
votes
2answers
244 views

Polynomial satisfying $ P \big(P (x)\big)=P (x)+ P\big(x^2\big)$

If $P(x)$ is a polynomial with integer coefficients such that for all integer $x$, $$P (P (x)) = P (x)+P (x^2).$$ I've tried solving it putting it as a function instead. Not much though. How do you ...
7
votes
3answers
175 views

Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$

I got this problem: Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$. (Hint: the solution involves limits at ...
9
votes
3answers
135 views

How find this all function $f(x^n+2f(y))=(f(x))^n+y+f(y)$

Question: Given a positive integer $n\ge 2$ . Find all functions $f:R\to R$, such that $$f(x^n+2f(y))=(f(x))^n+y+f(y)$$ let $x=0,y=0,a=f(0)$ then $$f(2f(0))=(f(0))^n+0+f(0)\Longrightarrow ...
9
votes
4answers
555 views

What functions satisfy this functional equation?

$$f(x)-g(x)=f(g(x))$$ How could I find an f(x) and g(x) that satisfy this?
5
votes
1answer
110 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
5
votes
1answer
87 views

Finding all real valued functions that satisfy $f(f(y) + xf(x)) = y + (f(x))^2$

I would like some help with finding all real valued functions that satisfy this equation: $f(f(y) + xf(x)) = y + (f(x))^2$ I tried the usual substitutions like $x = y = 0$, but my experience with ...
0
votes
0answers
32 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...
0
votes
1answer
43 views

Writing roots of f(x) as f(a) for some a

I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots ...
0
votes
0answers
36 views

Does there exist a solution of that system of functional equations?

Does there exist a non-constant rational function $f(x)$ (i. e. a ratio of two polynomials in $x$ over the reals) which simultaneously satisfies $f(x)=f(1-x)$ and $f(x)=f\left(\frac 1 x \right)$ on ...
5
votes
1answer
104 views

How find this function such $f(2010f(n)+1389)=1+1389+1389^2+1389^3+\cdots+1389^{2010}+n$

Question: Find all function: $f:N\to N$, such that $$f(2010f(n)+1389)=1+1389+1389^2+1389^3+\cdots+1389^{2010}+n,\forall n\in N$$ Maybe this is 2010 Mathematical olympiad problem.But I ...
2
votes
1answer
47 views

How to solve the following delay differential equation?

What is the solution for the following equation? $$\frac\partial{\partial q}f(s,q)= \frac s2 f(s+2,q)$$ Note, it is known that the solution for $$\frac\partial{\partial q}f(s,q)= s f(s+1,q)$$ ...
0
votes
0answers
20 views

Extract independent parameters

I have a two-variable function which depends also on a number of parameters (six to be exact) $f(x,y; c_1, c_2, c_3, \dots, c_6)$. The explicit form is quite complicated so I will not give it here. It ...
0
votes
1answer
15 views

How do I find and list compositions for (f) and (g)?

Ok, I've literally just spent the last 2 hours just to figure out two compositions problems for homework, and I've about had it. Anyone here that can help? Problem 1 $$ f(x) = 2x(2) - x -3 $$ $$ ...
1
vote
1answer
65 views

Solution of functional equation

i know the solutions of the well known Cauchy-functional-equation $f(x+y)=f(x)+f(y)$ But what does it change if i have the following form $f(x+g(y))=f(x)+f(g(y))$ ? what can i say about g? ...
0
votes
2answers
68 views

Cauchy's functional equation with polynomial

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function that satisfies $$ f(u+av)=f(u) + f(av) + P_n(u,v) $$ where $a$ is a known constant and $P_n(u,v)$ is a polynomial in $u$ and $v$ of ...
2
votes
2answers
63 views

Is there such a function $f:R\rightarrow R$?

Is there such a function $f:\mathbb R\rightarrow\mathbb R$, that for any real $x$ and $y$, we have the equality: $$ \frac{f(x)+f(y)}{2}=f\left({\frac{x+y}{2}}\right)+|x+y|\;\;\;? $$
2
votes
0answers
35 views

The existence of integral equations solution for a 2-dimensional unknown function

Suppose $f(\cdot,\cdot)\in C[0,1]^2$ is a kernel. $f$ is integratble $\int_0^1\int_0^1 f(x,y)dxdy<\infty$. $a,b\in C[0,1]$ are known functions, and $z(\cdot,\cdot)\in C[0,1]^2$ is a 2-dimensional ...
5
votes
0answers
170 views

Width of the Eiffel Tower as a function of height?

In the preface of Advanced Engineering Mathematics, 2nd Ed. by Zill and Cullen, it is claimed that the function relating the width of the Eiffel Tower as to the distance from its top, $x \mapsto ...
7
votes
1answer
479 views

How do you solve $f'(x) = f(f(x))$?

A friend told me to solve the following differential equation: $$f'(x)=f(f(x))$$ I have no idea how to solve this! This doesn't seem to be an ordinary differential equation and I can't even solve ...
2
votes
1answer
23 views

Characterizing functions which satisfy de Moivre's theorem

Let $f$ and $g$ be two non-zero functions $ \mathbb R \to \mathbb R $ which are continuous and differentiable everywhere. Furthermore, say that for all integer $n$: $$ (f(\theta) + i g(\theta))^n = ...
6
votes
2answers
196 views

Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$ 1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}} $$ EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a ...
1
vote
0answers
49 views

Solve the function from the composition [duplicate]

I have equations as follows $$f(f(x))=x^2+x$$ Then solve for $f(x)$. Can anyone give some hints about this question?
31
votes
8answers
1k views

How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?

How to get $f(x)$, if we know that $f(f(x))=x^2+x$? Is there an elementary function $f(x)$ that satisfies the equation?
3
votes
0answers
78 views

How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
0
votes
0answers
22 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...
0
votes
3answers
82 views

How to write a summation function that counts the number of nodes in a tree?

I come from a programming background and am interested in learning how to represent some things as simple equations, as an entry into thinking mathematically. How do you represent a tree structure as ...
3
votes
1answer
169 views

Solve for $f(x)$ if $f(f(x))=6x-f(x)$

If $f: [0,\infty) \rightarrow [0,\infty)$ and $f(f(x))=6x-f(x)$ $f(x)>0$ $ \forall x \in (0,\infty) $ Find f(x)
4
votes
0answers
247 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} ...
7
votes
4answers
104 views

Continuity of function where $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function which satisfies $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$ is continuous at $x=0$, then it is continuous at every point of $\mathbb{R}$. ...