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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36 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
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0answers
89 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 ...
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1answer
106 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
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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
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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 ...
1
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1answer
45 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 ...
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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 ...
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0answers
68 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 $. ...
0
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1answer
30 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
91 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?
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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 ...
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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
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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
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1answer
24 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
45 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
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1answer
40 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)) = ...
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2answers
78 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
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1answer
16 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 ...
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0answers
41 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 '' ...
6
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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 ...
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: ...
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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
58 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.
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2answers
68 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
75 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 ...
1
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2answers
49 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
93 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 ...
3
votes
2answers
56 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 ...
0
votes
1answer
29 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
21 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 ...
3
votes
1answer
65 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
57 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 $\varphi$.
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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
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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
58 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
80 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}, ...
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 ...
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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?. ...
2
votes
1answer
85 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 ...
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 ...
1
vote
3answers
43 views

A function that is zero for every integer multiple of $k$

I'm new to this kind of question so it may be a trivial one but I can't find a general solution: I'm searching for a function that has a value of $0$ for every integer multiple of some fixed real ...
5
votes
1answer
39 views

Find the function $f(x)$

Given $f(x)$ is a differentiable function such that $$f(x+y)=e^xf(y)+e^yf(x)$$ $\forall$ $x,y$ $\in$ $\mathbb{R}$ and $f'(0)=1$. Find $f(x)$ if we put $y=0$ we get $$f(x)=e^xf(0)+f(x)$$ $\implies$ ...
0
votes
0answers
40 views

Finding the “hidden values”.

I have this problem but my knowledge of the methods of linear algebra is totally useless if it exists at all (I know what is a matrix at least... maybe). I have a three collections of objects ${\bf ...
5
votes
2answers
131 views

Is there a function satisfying the following properties $ f^{n}(x)=(f(x))^{n+1}$??

Is there a function with the following properties? $$ f(x)=f(x) $$ $$ f'(x)=f(x)^2 $$ $$f^{(n)}(x)=\left(f(x)\right)^{n+1}$$ where $f^{(n)}$ denotes the $n$th derivative, and by convention ...
3
votes
3answers
172 views

if I know $f(x+1) = 2f(x) + 1$, how do I solve f(x)

This is just my thought on run time of a binary search: if you are allowed to make 1 comparison, you can search a sorted list of length 1, but if you are allowed to perform 2 comparisons, you can ...
0
votes
2answers
52 views

Functional equation $f(h(y)x+y)=g(y)f(x)$

(Note: this is a simplified version of my previous question, which was not answered). I am seeking the solution for the functional equation $f(h(y)x+y)=g(y)f(x)$ where $f,g,h$ are continuous. ...