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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19 views

Logarithmic functional equation on positive rational numbers

Let $\Bbb Q^+$ be the set of positive rationals and $f:\Bbb Q^+ \to \Bbb R$ be a continuous function satisfying $$ f(rs)=f(r)+f(s) $$ for all $r, s\in \Bbb Q^+$. Then $f$ has only the form $$ f(r)=c ...
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1answer
38 views

Multiplicative function on rationals [duplicate]

Let $\Bbb Q^+$ be the set of positive rational numbers. Find all solutions $f:\Bbb Q^+ \to \Bbb R$ of the functional equation $$ f(xy)=f(x)f(y), \quad x, y\in \Bbb Q. $$ Is $f(x)=x^a$ the only ...
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0answers
24 views

Solving differential functional equations with a restricted solution

I have a variable vector $X=\{x_1,x_2,...,x_n\}$, and a constant vector $V=\{v_1,v_2,...,v_n\}$. $f(x_i,X)$ is a function that takes X and xi as the parameter, for example: $f(x_i,X) = ...
4
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2answers
59 views

Solve functional equation [closed]

find all functions $f:\mathbb{R^{*}}\to \mathbb{R}$ such that $$f(y^2f(x)+x^2+y)=x(f(y))^2+f(y)+x^2,\;\forall x,y\in \mathbb{R^{*}}$$ ($\mathbb{R^{*}}=\{x\in\mathbb{R},x\ne 0\})$
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0answers
51 views

Solving functional equation

Problem:find all continuous functions $f:[0,+\infty)\to [0,+\infty)$ such that $$f(y^2f(x)+x^2f(y))=xy(f(x)+f(y)),\;\forall x,y\in [0,+\infty)$$
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1answer
42 views

Functional equation with $f(1)$ integer

Here is a nice problem: Let $f:R\rightarrow R$ be a function, R is the set of real numbers, satisfying the following properties: $ f(1)$ is an integer and $xf(y)+yf(x)=(f(x+y))^2-f(x^2)-f(y^2)$, for ...
4
votes
2answers
104 views

Functional Equation : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x).

Problem : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x). My approach : The given equation can be written as $$(x-y)f(x+y) -(x+y)f(x-y) =4xy(x-y)(x+y)$$ $$\Rightarrow ...
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1answer
33 views

Solution to a general scaling problem $G(\lambda z)=\frac{G(z)}{\gamma z^n}$

When playing with the scaling problem $$G(4z)=\frac{G(z)}{2z}$$ (see also this question) I discovered, that the general problem $$G(\lambda z)=\frac{G(z)}{\gamma z}$$ with two constants ...
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2answers
50 views

A scaling functional equation

I want to find a closed form of a function satisfying $$G(4z)=\frac{G(z)}{2z},$$ unfortunately I am not experienced with scaling problems, so I have no idea and would be thankful for any hints. ...
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1answer
71 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
141 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
43 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 ...
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0answers
51 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
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1answer
64 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, ...
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1answer
80 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
61 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 ...
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4answers
100 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
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1answer
52 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 ...
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2answers
79 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 ...
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1answer
82 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 ...
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2answers
115 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
45 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
123 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 + ...
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1answer
53 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 ...
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votes
3answers
89 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
62 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 $F(\omega)$ ...
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2answers
54 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 ...
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1answer
75 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 ...
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4answers
484 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
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2answers
268 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 $Q(X)=P(X^2-2) = (P(X))^2-2$ ...
27
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2answers
559 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
49 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
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0answers
22 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 ...
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0answers
21 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 = ...
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1answer
52 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
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1answer
139 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
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0answers
49 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 ...
3
votes
2answers
116 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
78 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
190 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 ...
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0answers
57 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
184 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
218 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
288 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
157 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
35 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
265 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
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1answer
55 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, ...