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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4
votes
2answers
102 views

Functional equation $x\space f(x^2) = f(x)$

How can I logically lead to the answer from the following conditions? $$ \left\{ \begin{align} & x \, f(x^2) = f(x) \text{ for all } x > 0, \\ & f(x) \text{ is continuous}, \\ &f(1) = ...
4
votes
3answers
142 views

Find $f$ if $ f(x)+f\left(\frac{1}{1-x}\right)=x $

Find $f$ if $$ f(x)+f\left(\frac{1}{1-x}\right)=x $$ I think, that I have to find x that $f(x) = f\left(\frac{1}{1-x}\right)$ I've tried to put x which make $x = \frac{1}{1 - x}$, but this equation ...
7
votes
1answer
105 views

Prove $f(x)=0$ when $f(2x^2-1)= f(x)\cdot 2x$

Prove that $f(x)=0$ for $$x\in[-1, 1]$$ $f-continuous$ and for all $x$: $$f(2x^2-1)= f(x)\cdot 2x$$ It is simple for integer numbers. Another fact that I've noticed that $$f(2(-x)^2-1)= f(-x)\cdot (-...
1
vote
0answers
36 views

Functional equation for $\sum_{n=1}^{\infty} \sinh(cn)^{-s}$?

Does anyone know of any kind of functional equation (or closed form) for $\sum\limits_{n=1}^{\infty}\sinh(cn)^{-s}$, where $c$ is an arbitrary constant? I've been messing around with it off and on for ...
0
votes
0answers
30 views

Time delay equation

If $x(t)=\displaystyle\frac{1+(x(t-T))^m}{(x(t-T)))^m}$ for all $t$ where $T$ is constant and $x(t)=x_s$ is the solution to the above equation, why can I write that: $x_s=\displaystyle\frac{1+{...
3
votes
1answer
55 views

Given $f(x+y)=f(x)+f(y)$ and $f(0)=0$, what can we say about f(x)

Given $f(x+y)=f(x)+f(y)$ and $f(0)=0$, what can we deduce about $f(x)$? I intend to say that $f(x)=x$, but find difficult to prove it. Is my guess correct, or wrong?
1
vote
3answers
85 views

Find all function satisfying $f(f(n))+f(n)=2n+3k$

Find all functions $f:\mathbb{N_{0}} \rightarrow \mathbb{N_{0}}$ satisfying the equation $f(f(n))+f(n)=2n+3k,$ for all & $n\in \mathbb{N_{0}}$, where $k$ is a fixed natural number. A friend of ...
0
votes
1answer
39 views

Integral functional equation.

$f(x)= \Big(\int _1^x g_1(t)g_2(t)dt\Big)\Big(\int _1^x g_3(t)g_4(t)dt\Big)-\Big(\int _1^x g_1(t)g_3(t)dt\Big)\Big(\int _1^x g_2(t)g_4(t)dt\Big) $ $\forall$ $x$ $\in R$ Where $g_1(x),g_2(x),g_3(x)$ ...
1
vote
0answers
90 views

Equidistant sets and extremal points

Let $\lVert\cdot \rVert$ be any norm on $\mathbb{R}^n$ and let $x\in \mathbb{R}^n$ be a point. Let $\textrm{ext}(B_{\lVert x\rVert})$ denote the set of all extremal points of $B_{\lVert x \rVert}:=\{...
0
votes
1answer
110 views

Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$? [duplicate]

Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$? In particular I'm interested in the qualitative properties of the such solutions.
1
vote
1answer
40 views

How to prove $X(t)$ is differentiable?

Suppose $X(t)\in M_n(\Bbb R), t\in\Bbb R$ and is continuous, invertible at every point on the real line, if the equation $$X(t)X(s)=X(t+s)$$ holds for all $t,s\in\Bbb R$, prove that there exists a ...
2
votes
0answers
25 views

Show that retarded argument functional operator is surjective

I want to show that for every $\lambda \neq 0$ and $g\in L^{2}(\mathbb{R})$, there is a $f \in L^{2}(\mathbb{R})$ such that $$e^{-|x+1|}f(x) - \lambda f(x+1) = g(x)$$ I tried looking at the operator ...
2
votes
1answer
49 views

Find all real polynomials $P(x)$ which satisfy the equation$ P(x)P(-x)=P(x^2-1)$

Find all real polynomials $P(x)$ having only real zeros and which satisfy the equation $$P(x)P(-x)=P(x^2-1)$$ Please explain me the process and refer some books to learn polynomials. Thanks ...
0
votes
0answers
19 views

How to convert $\ max \int_{-\infty}^{\infty} f(x) dx =\ min \int_{-\infty}^{\infty} g(x) dx $

Let $f$ or $g$ have a given condition. ( example $\int_{- \infty}^{\infty} f(x) exp(-x^2) dx = 1$ Or $g(g(x)) = x^3$ Or a differential equation for one of $f,g$. ) I want to find a general way - if ...
1
vote
0answers
69 views

Minimizing $\int_{0}^{1} (1+x^2)f(x)^2 dx$ for $f(f(x)) = x^2$

What is $$\min_{f\in D} \int_{0}^{1} (1+x^2)f(x)^2\mathrm dx,$$ where $D$ is the collection of all continuous real functions from $[0,1]$ such that in the interval $[0,1]$ we have $f(f(x)) = x^2 $. ...
-1
votes
1answer
31 views

Functional equaliton on continuous function $f:[0,1]\to \mathbb{R}$

Let $f:[0,1]\to \mathbb{R}$ be continuous, and suppose that $f(0)=f(1).$ Show that there is a value $x\in [0, 1998/1999]$ satisfying $f(x)=f(x+1/1999)$.
6
votes
1answer
93 views

How can I find $f(3)$ if $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$?

If $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$ and $f(1) = 2$ then find $f(3)$. Can you give me any hint that I can start with?
1
vote
2answers
70 views

Express $y$ from $\ln(x)+3\ln(y) = y$

i am finding a inverse function to $$y=\frac{e^\frac{x^3}{3}}{x^3}$$ First step: $$x=\frac{e^\frac{y^3}{3}}{y^3}$$ Next: ... $$(xy^3)^3 = e^{{y}^3}$$ and now in this step i have no idea how can i ...
0
votes
0answers
22 views

Existence of an inverse function in this functional equation

Let $V,W$ each be connected and separable sets. Suppose we have continuous functions $F:\mathbb{R}\times V\rightarrow \mathbb{R}$, $G:\mathbb{R}\times V\rightarrow \mathbb{R}$, $f:V\times W\rightarrow ...
2
votes
1answer
25 views

Functional inequality $f(y)+xf(x)≤yf(x)+f(f(x))$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function sucht that: $$ f(y)+xf(x)≤yf(x)+f(f(x)) $$ for all $x,y\in\mathbb{R}$. Show that $$ f(x)+yf(x+y)≤0 $$ for all $x,y\in\mathbb{R}$. I tried some ...
2
votes
1answer
30 views

Cauchy's functional equation for involutions

It is well known that Cauchy's functional equation for $f:\mathbb{R}\to\mathbb{R}$ $$ f(x+y)=f(x)+f(y)\space\forall x,y\in\mathbb{R} $$ admits highly pathological solutions if no further conditions ...
2
votes
1answer
46 views

Functional equation $f(x^3) - f(x^3 - 2) = (f(x))^2 + 12$

$$f(x^3) - f(x^3 - 2) = (f(x))^2 + 12$$ Given the functional equation above, I am trying to find the value of $f(3)$. I do not remember the exact statement of the problem precisely, so I am not sure ...
1
vote
1answer
44 views

Solve the functional equation (medium-hard)

Find all real functions $f(x)$ $\mathbb{R} \to \mathbb{R}$ such that $x^2f(yf(x)) = y^2f(x)f(f(x))$ Obviously, let $y=0$ you instantly get, $f(0) = 0$. Also, a relation is: $f(yf(x)) = \frac{y^2 ...
0
votes
2answers
80 views

How can one show that$f(x^n)=nf(x)$

How can one show that if $f(xy)=f(x)+f(y)$ holds for all for real $x$ and $y$ that $$f(x^n)=nf(x).$$ How can i prove that $f(\frac{1}{x})=-f(x)$ To calculate f(1) do i need to pute x=1 ? Do i need ...
1
vote
1answer
18 views

Determine formula to calculate inverse range between two numbers

I've got a measurement for a graphics processing filter that increases intensity as the value decreases. For whatever reason, this filter's default value (minimum) is 1.75, and I've set the maximum ...
0
votes
0answers
43 views

Uniqueness question from functional equation

Let $f(x)$ and $f_2(x)$ be real and continuous on the interval $[0,\infty[$. Let $f(1) = f_2(1) = 1 , f(2) = f_2(2) = e$. Let $g(x)-g(x-1) = f(x-1)$. Let $ f ' (x) = exp(g(x) - g(1)).$ and $f '' (1)...
6
votes
1answer
96 views

How to prove a function is periodic from a given functional equation?

Given that $f: \mathbb R \to \mathbb R$, and that for some $a \in \mathbb R$, $f(x+a)={1\over 2}+\sqrt{f(x)-f^3(x)}$; prove $f(x)$ is a periodic fucntion. I know that to prove a function is periodic ...
0
votes
1answer
16 views

Re-arrangement of equation

This type of question might be voted down or frowned upon, but if anyone could point me in the right direction I would be very grateful. I have worked through a question getting the following result: ...
0
votes
0answers
13 views

Formulation of a rule

Im new to this forum(hence, formatting issues). I come from CS background. I am trying to figure out a formula to code set of rules. The requirement is as below. Imagine two parameters X and Y ...
3
votes
3answers
59 views

If $g(x) = \frac{x}{x+1}$, and $f\circ g(x) = x^2$, find $f(x)$? [closed]

$$g(x) = \frac{x}{x+1}$$ $$f\circ g(x) = x^2$$ $$f(x) =\text{ ?}$$ Hello, I don't have any clue how to solve that. Do you have any ideas how to solve that? Thanks in advance.
-1
votes
2answers
69 views

Solve the functional equation $f(x) + f(x^2) = 2$ [closed]

What are the solutions of the functional equation $f(x) + f(x^2) = 2$? Will they be one to one or many to one? Will they be periodic or not?
3
votes
1answer
69 views

Polynimials $P(x)$ and $Q(x)$ satisfying $P(x^2+1)=(Q(x))^2+2x, Q(x^2+1)=(P(x))^2$

I am looking for all polynomials $P(x)$ and $Q(x)$ satisfying the simultaneous functional equations given in the title, i.e. $P(x^2+1)=(Q(x))^2+2x$ $Q(x^2+1)=(P(x))^2$ I've already found one ...
3
votes
2answers
67 views

Can one do anything useful with a functional equation like $g(x^2) = \frac{4x^2-1}{2x^2+1}g(x)$?

I got $$g(x^2) = \frac{4x^2-1}{2x^2+1}g(x)$$ as a functional equation for a generating function. Is there a way to get a closed form or some asymptotic information about the Taylor coefficients ...
3
votes
1answer
87 views

Extremal of a functional $I=\int\limits_0^{x_1} y^2(y')^2dx$

The extremal of the function $$I=\int\limits_0^{x_1} y^2(y')^2dx$$ that passes through $(0,0)$ and $(x_1,y_1)$ is a constant function a linear function of x part of a parabola part of an ...
1
vote
2answers
50 views

Can any functions be expressed in one formula? (without conditions)

This is the extension of my previous inquiry: Is it possible to describe the Collatz function in one formula? Can each of all functions be expressed in one formula? That is, can any function ...
3
votes
2answers
95 views

Solving $f(x)=f(\frac{1}{x}),\,\,\,x>0$

What are the non-trivial solutions of the following functional equations: $$f(x)=f(\frac{1}{x}),\,\,\,x>0$$ given $f$ if differentiable on $(0,\infty)$. $\textbf{Edit:}$ Do all the solutions ...
4
votes
2answers
57 views

$f: [0,1] \to [0,1]$ is injective & $ f(2x-f(x))=x$ for $x \in [0,1]$ then $f(x)=x $

Let $f: [0,1] \to [0,1]$ be an injective function such that $ f(2x-f(x))=x$ for $x \in [0,1]$. I have to prove that $f(x)=x, x\in [0,1]$. My attempt: let $$g(x)=2x-f(x)$$ then $$f(g(x))=x \implies f$...
0
votes
1answer
30 views

Functional Equation Different Substitution Different Result

Okay Consider this g(x).g(y)=1 for all x,y ∈ ℝ Substitute x=y we get g(x)= 1 or -1 .Now Substitute y=1/x g(x).g(1/x)=1 we get g(x)=x^n where n∈N Different substitutions produced different ...
0
votes
1answer
22 views

Cauchy functional equation three variables

If I have function from $R^3$ to $R$ satisfying $f(x_1,x_2,x_3)+f(y_1,y_2,y_3) = f(x_1+y_1,x_1+y_2,x_3+y_3)$ is it necessarily linear? $f(z_1,z_2,z_3) = \lambda _1 z_1+\lambda _2 z_2+\lambda _3 z_3$...
3
votes
1answer
71 views

How to solve a functional differential equation?

$$(1) \quad \cfrac{d}{dx} (f(x^n))=\cfrac{-f(x^n)^2}{f(n \cdot x^{n-1})}$$ How do I solve this functional differential equation? I need a closed form solution, so approximations won't cut it, I'll ...
2
votes
2answers
59 views

Find the period of a function $\varphi:\mathbb{R}\backslash\{3\}\to \mathbb{R}$ [closed]

Let $\varphi:\mathbb{R}\backslash\{3\}\to \mathbb{R}$ a periodic function so that forall $x\in \mathbb{R}$ $$\varphi(x+4)=\frac{\varphi(x)-5}{\varphi(x)-3}$$ Find the period the $\varphi$.
1
vote
1answer
26 views

How to state a recurrence equation

I'm working on my homework and I saw this problem: A divide and conquer algorithm $X$ divides any given problem instance into exactly two smaller problem instances and solve them recursively. ...
0
votes
2answers
124 views

why this function is even function

Let $f(x)$ be a continuous differentiable odd function, and for any $x,y\in R$, such $$f(x-y)=f(x)g(y)-g(x)f(y)$$ show that $g(x)$ is an even function. How can I go about solving this? I ...
0
votes
2answers
37 views

Show that $y^n+1/y^n \in \mathbb{N}$

Let $y$ be the solution of the equation $x+\frac{1}{x}=3$ such that $y>1$. Show that $y^n+\frac{1}{y^n} \in \mathbb{N}$ We have $x+\frac{1}{x}=3$ if and only if $ x^2-3x+1=0$. Which has two ...
1
vote
1answer
59 views

Monotonically increasing functions 123

If $f$ is a monotonically increasing function and if $$f\left({x+f(x)\over2}\right)=x$$ for every $x\in\mathbb R$, prove that $$f(x)=x$$
2
votes
1answer
77 views

$F(F(x)+x)^k)=(F(x)+x)^2-x$

I have no idea about this problem. But I feel we have to use chain rule of differentiation here. The Function $F(x)$ is defined by the following identity: $F(F(x)+x)^k)=(F(x)+x)^2-x$ The value of $...
10
votes
1answer
81 views

Vector Space Structures over ($\mathbb{R}$,+)

Consider the abelian group ($\mathbb{R}$,+) of real numbers with the usual addition. Is there a scalar multiplication \begin{equation} \cdot : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, \end{...
2
votes
1answer
23 views

Two variable equation, weighting of stock

I am trying to find the weighting of stocks in a portfolio (both variables should be between 0 and 1 but added together should be 1). the equation is: $1.2353 = x(1.2) + y(0.9)$ I have it simplified ...
1
vote
4answers
149 views

How can I find $f'(2x)?$

it might seem a little bit elementary. $f$ is defined on $\Bbb R$ and it is differantiable. and is not equal to zero. if $xf(x)-yf(y)=(x-y)f(x+y)$ then find what is $f'(2x)$ equal to?. $(f'(x),2f'(x),...
2
votes
1answer
88 views

For which polynomials $P$ does there exist a polynomial $Q$ such that $P(x)=Q(x)Q'(x)$?

I am wondering for which polynomials $P$ does there exist a polynomial $Q$ such that $P(x)=Q(x)Q'(x)$? I am quite sure a complete characterisation will be very hard, but I'm looking for partial ...