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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1answer
66 views

Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$? [duplicate]

Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$?
5
votes
5answers
136 views

$\forall x\in\mathbb R$, $|x|\neq 1$ it is known that $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$. Find $f(x)$.

$\forall x\in\mathbb R$, $|x|\neq 1$ $$f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$$Find $f(x)$. Now what I'm actually looking for is an explanation of a solution to this ...
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0answers
31 views

Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
0
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0answers
43 views

Does there exist a nontrivial cumulative distribution function $F$ on $\mathbb{R}_+$ for which $(F^2)'= (F')^{\ast k}$ for some $k > 0$?

This question arises from the context of computing the distribution of total execution time with an underlying graph of tree. For instance, we can model the execution times of all the nodes on the ...
3
votes
1answer
55 views

Functional inequality and one identity

I'm a high school student from Bonn, Germany and I have to solve the following problem: If $g:R \rightarrow R $ is a function with the property $g(ab)-ag(b)\leq bg(a)$, for all real numbers a and b, ...
1
vote
1answer
54 views

Find $g(x)$ if $f(g(x))=f(x)g(x)$ and $g(2)$=37, $f(x)$ and $g(x)$ are polynomials

Suppose $f(x)$ and $g(x)$ are non-zero polynomials with real coefficients, such that $f(g(x))=f(x)\times g(x)$. If $g(2)=37$, find $g(x)$. I tried plugging $f(x)$ and $g(x)$ as $n$ and $m$ ...
4
votes
1answer
51 views

Find all function satisfying a condition with $\min$ and $\max$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$ \forall (x,y)\in\mathbb{R}^2,x\ne y,\quad \min (f(x),f(y)) \leq \frac{f(x)-f(y)}{x-y} \leq \max(f(x),f(y)) $$ I have started with ...
0
votes
4answers
85 views

Functional equations

Let $f:\mathbb{R}\to \mathbb{R}$ is a function such that for all real $x$ and $y$, $f(x+y)= f(x) + f(y)$ and $f(xy)= f(x)f(y)$, then prove that $f$ must be one of the two following functions: ...
2
votes
1answer
41 views

Functional equation of non-negative function

Find all $ f:[0,\infty)\rightarrow [0,\infty) $ such that $ f (2)=0 $, $ f (x)\not= 0 $ for $ x\in [0, 2) $ and $$ f (xf (y)) f (y)=f (x+y) $$ for all $ x, y\ge 0 $. I tried plugging in values ...
7
votes
2answers
67 views

Non trivial solutions of $g\circ f-f\circ g=g\circ f\circ g$

While thinking of perfect numbers, I came across the functional equation $g\circ f-f\circ g=g\circ f\circ g$ where the unknowns $f$ and $g$ are functions from $\mathbb{R}$ to itself. I only know one ...
1
vote
1answer
78 views

A Funtional Equation

Find all functions ${\rm f}:{\mathbb N}\times{\mathbb N} \rightarrow {\mathbb N}$ satisfying $$ \begin{array}{rrcl} a) & {\rm f}\left(n,n\right) & = & n \\[2mm] b) & {\rm ...
1
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2answers
87 views

On a function with a (complicated) functional equation.

Let $g(x,y)$ be a function such that: I. $-1\lt g(x,y)\lt1.$ II. $$\ln(\frac{1+g(x,y)}{1-g(x,y)})+2y\tan^{-1}(yg(x,y))=2(y^2+1)x,$$ for $x\in\mathbb R, y\gt1.$ Then i. Show that $g(x,y)$ ...
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1answer
43 views

What is the family of functions that satisfies

Let $f$ be a continuous function in $\Re$, differentiable (at least one time). Then, which functions can satisfy: $$ \frac {f(x)} {f'(x)}= - \frac {f(x-\frac {f(x)} {f'(x)})}{f'(x-\frac {f(x)} ...
4
votes
2answers
113 views

Find all functions ${\rm f} :{ \mathbb R}_{+}\to{ \mathbb R}_{+}$

Find all functions ${\rm f}:{\mathbb R}_{+} \to {\mathbb R}_{+}$ , such that $\forall\ x,y \in \mathbb R_+$ the equation $$ \left[1 + y{\rm f}\left(x\right)\right]\left[1 - y{\rm f}\left(x + ...
3
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1answer
51 views

Solutions to $g(ab) = ag(b) + bg(a)$ - “Zero function question”

This question was recently asked and then voluntarily removed by its author. I thought it was interesting enough to get an answer. The question was as follows: "The function $g:\mathbb R\to\mathbb ...
3
votes
3answers
82 views

Functional Equation : $f(x) = f(x + y^2 + f(y))$

This problem is from my textbook: Given : $f:\mathbb R\to\mathbb R$ Solve this functional equation : $f(x) = f(x + y^2 + f(y))$ I think this function is just a simple constant, so I try all my ...
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1answer
18 views

Finding Inverse of Function With Two Instances of X

I need to find $f^{-1}(2)$ where $f(x) = 2 + x^2 + tan(πx/2)$ I know can substitute $f(x)$ with $y$ and swap $x$ and $y$: $$x = 2 + y^2 + tan(πy/2)$$ But I'm having trouble eliminating the tangent: ...
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0answers
36 views

Fourier transform of a function involving $\sec(\omega)$

The summary of my question is: What should I make of $\mathcal{F}^{-1}\left[\frac{\csc(\omega)}{\omega^2-\beta^2}\right]$ where $\mathcal{F}^{-1}$ is the inverse fourier transform (taking $F(\omega)$ ...
4
votes
2answers
34 views

How to take partial derivatives of functions whose inputs depend on the same variable?

I am starting to learn about the Calculus of Variations and the Euler-Lagrange equation is extremely confusing to me: The Euler–Lagrange equation, then, is given by ...
1
vote
1answer
65 views

A fairly difficult differential equation

I was thinking what if I had a differential equation of the form: $$\frac{d^2u}{dx^2} + vu(x) = 0 $$ where $v(y(x))$, that is $y$ is a function of $x$. What are the possible restrictions that I can ...
12
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4answers
420 views

Find all bijections $\,\,f:[0,1]\rightarrow[0,1]\,$ satisfying $\,\,f\big(2x-f(x)\big)=x$.

A friend of mine gave me the following problem: Find all functions $f:[0,1]\to[0,1]$, which are one-to-one and onto and satisfy the functional relation $$ f\big(2x-f(x)\big)=x, \tag{1} $$ for all ...
8
votes
2answers
239 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 $Q(X)=P(X^2-2) = (P(X))^2-2$ ...
25
votes
2answers
450 views

How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.

Prove that there is no continuous functions $f:[0,1]\to R$ such that $$ f(x)+f(x^2)=x,x\in [0,1]. $$ My try: Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ ...
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1answer
48 views

When can an arbitrary function be put into this form?

A problem I've been trying to address but not been able to get very far with is to devise a method to check whether or not some function $q(x)$ can be written like $$ q(x) = ...
0
votes
0answers
19 views

Linear-like function

Suppose we need to find all the functions $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that $\forall x,y,z\in \mathbb{R} ~~ f(x+z) - f(x) = f(y+z) - f(y)$ It can be shown that $~~\forall r \in ...
0
votes
0answers
16 views

Mellin transform and a proof of the functional equaton for the $ \zeta (s) $

i would like to obtain a proof of the functional equation for the RIemann zeta function to do so i would like to know if there is a function or a distribution so $$ \int_{0}^{\infty}f(t)t^{s-1}dt = ...
0
votes
1answer
39 views

Worded Problem: Model a Plane Landing

I have no idea to model this. All I know are the two points $(50, 10)$ and $(0,0)$ Then from after solving I get $a=1/12500$ and $b=0$ The textbook answers are:
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1answer
121 views

What is the solution of the following functional equation? (I must confess it is a headache for me)

Find all the functions $f: \mathbb{Z} \to \mathbb{Q} $ such that $f(\frac{x+y}{3})=\frac{f(x)+f(y)}{2}$; $\forall x,y\in\mathbb{Z}$ knowing that $\frac{x+y}{3}\in\mathbb{Z}$.
3
votes
1answer
114 views

Functional equations leading to sine and cosine

This question is a possibly harder version of: Find $g'(x)$ at $x=0$. Question. Let $f,g :\mathbb R\to\mathbb R$, such that \begin{align} f(x-y)=f(x)\, g(y)-f(y)\, g(x), \tag{1}\\ g(x-y)=g(x)\, ...
0
votes
0answers
45 views

Non-constant solution of $f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$ in rings

The answers to this question prove that over an integral domain the functional equation $$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$ has only solutions where one of $f$ or $g$ is constant. The proofs make ...
2
votes
2answers
112 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 ...
2
votes
1answer
76 views

Functional Equation $f(n) = 2 f(n / f(n) )$

I have the following functional equation: $f(n) = 2 \cdot f\left(\frac{n}{f(n)}\right)$ Under the precondition that $ f(n) = \omega(1) $, monotinic and a initial value $ f(1) = \Theta(1) $ one can ...
5
votes
3answers
173 views

Continuous solutions of $f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$

Consider the following functional equation: $$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$ where the equation holds for all $x,y,z \in \mathbb{R}$. One solution is $f(x)=cx$ and $g(x)=1$. What are all the ...
0
votes
0answers
35 views

Probability distribution satisfying constraints?

Continuing from this question. Given two random variables $X$ and $Y$ where $X \sim \operatorname{Beta}(a, b)$ and $Y \sim \operatorname{Beta}(c, d)$, I'm looking for a random variable $Z$ with a ...
7
votes
4answers
145 views

Find all real to real function satisfy this functional equation.! $f((x+y)/(x-y))=[f(x)+f(y)]/[f(x)-f(y)]$

Find all real to real function satisfy this functional equation.! $$f\left(\frac {x+y}{x-y}\right)=\frac {f(x)+f(y)}{f(x)-f(y)}$$ I couldn't get to the final answer but I get $f(0) = 0$ and $f(1) = ...
7
votes
1answer
203 views

Find all continuous functions $ f:\mathbb{R} \rightarrow \mathbb{R}$ such that

Find all continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x)^2 - 2\Bigl\lfloor\,2\,\bigl|f(2x)\bigr|\,\Bigr\rfloor f(x) + \Bigl\lfloor{4f(2x)^2-1}\Bigr\rfloor = 0\,.$$ Here ...
3
votes
2answers
243 views

Find a solution for f(1/x)+f(1+x)=x

Title says all. If f is an analytic function on the real line, and $f(1/x)+f(x+1)=x$, what, if any, is a possible solution for $f(x)$? Additionally, what are any solutions for $f(1/x)-f(x+1)=x$?
8
votes
2answers
139 views

How find this function equation $(f(x))^2-(f(y))^2=f(x+y)\cdot f(x-y)$

Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that satisfying the function equation $$(f(x))^2-(f(y))^2=f(x+y)\cdot f(x-y)$$ By the way :I have see this problem( is ...
0
votes
0answers
29 views

Symmetric expressions for homogeneous functions

Suppose $f(x_1, \ldots, x_n)$ is a homogeneous function, i.e. a function such that $$f(\lambda x_1, \ldots, \lambda x_n) = \lambda^d f(x_1, \ldots, x_n)$$ for all $\lambda$ and for some positive ...
6
votes
3answers
256 views

How to solve this infinite set of equations?

If I can find a solution to the following set of equations then, with a bit of luck, I should be able to derive all sorts of nifty new results in non-equilibrium statistical mechanics. However, I'm ...
1
vote
1answer
54 views

Reference request: Where is this functional equation found?

$$ g\left(\frac{x+y}{1+xy}\right) = g(x)g(y). $$ One solution is $$ g(x) = \frac{1+x}{1-x}. $$ Another is $$ g(x) = \sqrt\frac{1+x}{1-x}. $$ Any other power of the first solution is also a solution, ...
7
votes
1answer
97 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 ...
2
votes
1answer
45 views

Existence of a function with a changing period

$f,\alpha$ are continuous $\mathbb{R}\to\mathbb{R}$ functions satisfying: $$f\big(x+\alpha(x)\big)=f(x)$$ If $f$ is non-constant, must $\alpha$ be constant? My idea was to use the fact ...
0
votes
0answers
22 views

Question on 2 functional equations.

Let $z,x$ be complex numbers. Im looking for analytic functions $f(z)$ such that : $$1) \exp(\ln^{5} (f(x))=\sum_i a_i f(b_ix)$$ $$2)f(x)^5=\sum_j c_j f(d_jx)$$ holds for all $x$ and where both ...
2
votes
2answers
36 views

Find all functions

Hi I'm not sure if I'm correct in this example: Find all functions such that $f(x-|x|)+f(x+|x|)=x$ where $x \in R$, so my answer is the are only one function satisfying this condition ...
3
votes
1answer
40 views

Functional equation with function defined on $\mathbb{N}^{*}$

Let $ f:\mathbb{N}^{*} \mapsto \mathbb{N}^{*} $ be a function with the following property: $$ \frac{f(x+1)f(x)-2x}{f(x)}=\frac{2f^2(x)}{x+f(x)}-1$$ Determine all functions with this property. (I'm ...
4
votes
5answers
145 views

Is there an everywhere-defined function that satisfies $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$

Is there a function $f:\mathbb{R}\to\mathbb{R}$ which is differentiable and satisfies the following: (1) $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$ (2) $f'(0)=1$ (1) is the functional equation for ...
0
votes
0answers
30 views

What is the analytical solution to a Volterra integral equation?

I need to solve a following equation: \begin{equation} r_{k+1} = -\sum\limits_{l=0}^{k-1} r_l \cdot (k-l) \cdot \left(\frac{\omega}{t_c - l}\right)^{2 \beta} + \delta_{k,0} \end{equation} subject to ...
1
vote
1answer
103 views

My proof of Cauchy functional equation?

Although I have not quite studied functional equations, I came upon Cauchy functional equation and tried to prove it. Here is what I have done: We are given the condition, $f(x+y)=f(x)+f(y)$. ...
3
votes
1answer
145 views

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

What are the real analytic functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...