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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6
votes
4answers
67 views

Find all polynomials with real coefficients that satisfy $(x^2-6x+8)P(x)=(x^2+2x)P(x-2)$

Find all polynomials with real coefficients that satisfy $$(x^2-6x+8)P(x)=(x^2+2x)P(x-2)\forall x\in\Bbb R$$ My work; $$\frac{P(x)}{P(x-2)}=-\frac{4}{x-2}+\frac{12}{x-4}+1\tag{1}$$ ...
4
votes
1answer
219 views

About an integral equation

I would like to obtain $g$ by solving the following integral equation $$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$ where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+ ...
16
votes
2answers
807 views

A functional equation with no solution

Let $f:\mathbb{R}\to (0,\infty)$ be a differentiable function satisfying $$f(f(x))=f^\prime(x)$$for each $x$. Show no such function exists. I got this problem in an exam. I haven't done anything ...
0
votes
3answers
53 views

functional equation for $x^2$ $f(f(x))=x^4$

If $f(f(x))=x^4$ for all real $x$ and $f(1)=1$ find $f(0)$. It seems that $f(x)=x^2$ but can we solve without this explicit form of $f$?
0
votes
0answers
117 views

An Interesting Function

What would be the fastest method to compute Hyperfactorial Function written below F(n,r)=H(N)/H(r)*H(N-r) where r < N where H(N)=(1^1)(2^2)(3^3).....(N^N)
3
votes
4answers
91 views

Functional equation $f(x)-f(y)=\frac{1}{(x-y)^{2}}$

Please help me to solve the functional equation $$ f(x)-f(y)=\frac{1}{(x-y)^{2}} $$ for all real $x\neq y$. I have reduced it to $$ f(x+h)-f(x)=\frac{1}{h^{2}} $$ for all real $h\neq 0$. But what to ...
1
vote
0answers
139 views

Solving an integral (or series) equations system

Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ...
0
votes
0answers
18 views

How to create a equation that is linear or exponential taking account to the sum?

Example I want to sell 1000 products on 10 days. I want the development on sales to be linear. How do I create a equation that can tell me how much I should sell per day? if the days where the x ...
1
vote
3answers
73 views

If $f$ is continuous, $f(1) >1$ and $f(x+y)=f(x)f(y)$, then $f$ is increasing.

Consider the function $f$ with the following properties: $$\lim_{x\rightarrow 0} f(x) =1,$$ $$f(x+y)=f(x)\,f(y),$$ $$f(x) >0,\quad \forall x\in\mathbb{R},$$ $$ -\infty<x,y<\infty.$$ Show ...
3
votes
2answers
95 views

How to derive an proof for this infinite square root equation?

Here is continuous square root, namely: $\sqrt {1 + a \sqrt {1+b \sqrt {1+c\sqrt {1 +...}}}}$= any integer Find $a,b,c,d,e,f,...$ in general Uh, very interesting algebra pre-calculus problem, yet ...
2
votes
1answer
44 views

Functional-equation for strictly increasing functions

Let $n,c$-given natural numbers .Let $f(x)$ - strictly increasing function , domain of definition and set of values of ​​which are non-negative integers ,$f(0)=0 , f(1)=c$ and \begin{align} ...
2
votes
7answers
128 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
119 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
68 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
49 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
68 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
43 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
73 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
27 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 ...
0
votes
0answers
24 views

mathematical models

$S=S_{0}.\displaystyle\frac{4ae^{1/2d}}{(1+a)^2\cdot e^{a/2d}-(1-a)^2\cdot e^{-a/2d}} $ How do I find k in the above formula knowing that where $a=\sqrt{1+4ktd}$ ? Note: try to isolate k in the ...
1
vote
2answers
137 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
26 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
208 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
71 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*} ...
2
votes
2answers
73 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
84 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 ...
0
votes
0answers
34 views

What is this function of 2 variables?

Can you tell me the function f(K,N) that has the following values? For my education, please also explain how you tackled the problem. ...
1
vote
2answers
67 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
43 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
45 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
18 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
113 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
237 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
4answers
165 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
127 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
546 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
106 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
71 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
28 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
29 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
98 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
45 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
15 views

Extract independent paramets

I have a 2-variable function which depends also on a number of parameters (6 to be exact) $f(x,y; c1, c2, c3, .. c6)$. The explicit form is quite complicated so I will not give it here. It suffices to ...
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
58 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
66 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
61 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: $$ \frac{f(x)+f(y)}{2}=f\left({\frac{x+y}{2}}\right)+|x+y|\;\;\;? $$