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Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (...

3
votes
1answer
28 views

Mean of two i.i.d. random variables follows the same distribution, Cauchy distribution?

It is well known that if $X$ and $Y$ follow i.i.d. Cauchy distribution of scale $\gamma$, say $$ p_{\gamma} (x) = \frac{1}{ \pi \gamma ( 1 + x^2 / \gamma^2 ) }, $$ then their arithmetic mean $ ( X + Y ...
6
votes
1answer
92 views

Solving $f(x) = f(x/2) + f(x/3) + x$

How would one proceed to proving that the solution to the functional equation $$f(x) = f(x/2) + f(x/3) + x$$ is $f(x)=6x$ which is also unique? To clarify, I am not aware of neither the proof of the ...
3
votes
1answer
49 views

Solutions to $-t\ f'(t)+(1+r t)\ f(t)=f(t/2)$

Any thoughts on how to tackle this ODE $-t\ f'(t)+(1+r t)\ f(t)=f(t/2)$, $t\in[0,t_1]$, with boundary conditions $f(0)=0$ and $f(t_1)=s$, with $r,s$ and $t_1$ fixed constant.
0
votes
1answer
27 views

How can I solve distributed delay differential equation? [on hold]

I would like to solve the next unbounded distributed delay differential equation with Matlab $$\dot{x}=a x + \int_{0}^{\infty} f(t-s) ds$$
0
votes
1answer
32 views

Solving functional equation by comparising sides

I have a brain lag. I'm overthinking probably. I have the following equation: $$ f^{-1}(f(x)+f(y))=(g\circ h)^{-1}((g\circ h)(x)+(g\circ h)(y)), $$ where, say, $f,g,h$ are automorphisms on $\mathbb{R}$...
1
vote
1answer
18 views

Proof that solutions to Cauchy F.E. over $\mathbb{Q}$ are linear

I would like to prove that the solutions to Cauchy's functional equation over $\mathbb{Q}$ are linear, that is, all solutions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ have the property $f(x)=cx$ for some $...
-1
votes
0answers
46 views

Two functions on $\mathbb N$ such that $a(f(x))=x$ and $a'(f(x))=x$? [on hold]

Given function $f:\mathbb N→\mathbb N:x→x^2$, how does one find two functions $a:\mathbb N→\mathbb N$ and $a':\mathbb N→\mathbb N$, such that $a(f(x))=x$ and $a'(f(x))=x$? N being the set of natural ...
2
votes
1answer
58 views

Solving functional equation $\sin f = f^2-3if+\pi$

I want to find all (entire) solutions to the equation $$\sin f = f^2-3if+\pi.$$ Using the identity theorem, I was able to show that if a solution exists, it must be constant. Therefore, all that is ...
1
vote
0answers
34 views

The functional equation $f(-x+b)=f(x)$

I can solve the (periodic) functional equation $f(x+b)=f(x)$ completely ($x\in \mathbb{R}$ and $b\neq 0$). Indeed, its general solution is $f=\phi o (\; )_b$, where $(\; )_b$ is the $b$-decimal (...
2
votes
3answers
50 views

Let $f$ be a function $f : \mathbb{N} \to \mathbb{N}$ such that $f(2x+3y)=f(x)f(y)$, determine $f$

here what I did . $$f(0)=f(0)^2$$ so $f(0)=1$ or $f(0)=0$ IF $f(0)=1$ we have $f(2y)=f(y)$ $$f(1)=f(2)=\ldots=f(2^n)=a$$ the equation $f(x)-a=0$ has infinitly many solutions , so $f(x)=a$ since f(0)=...
2
votes
1answer
45 views

find all the functions $f:\mathbb{R}→\mathbb{R}$ such that $f(1)=1$ and $f(xy+f(x))=xf(y)+f(x)$

here is what i did i found that $f(f(0))=f(0)$ i need to prove that f is injective so that $f(0)$ will be equal to $0$ Hence $f(f(x))=xf(0)+f(x)$ so if $f(0)=0$ and $f$ is injective $f(x)=x$ please ...
1
vote
2answers
41 views

Continuity proof using $f(x+y) = f(x) + f(y)$ ($3$ parts, last part!)

Suppose that $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfies $f(x+y) = f(x) + f(y)$ for each $x, y \in \mathbb{R}$. $1)$ I've proven that $f(qx)=qf(x)$ for all $x \in \mathbb{R}$ and $n \in \mathbb{...
0
votes
3answers
74 views

Functional Equation: $f(x\cdot y)=f(x)+f(y)~~~\forall x, y>0$ [closed]

So I have the question: $$f(x\cdot y)=f(x)+f(y)~~~\forall x, y>0$$ We haven't learn these in class and I'm assuming it has a logarithmic answer. Could anyone please help?
0
votes
1answer
29 views

Method for solution to a recurrence

Is there a closed form solution or tight bound to recurrence $T[n]=k\cdot T[n^{1/c}] + (\log n)^{r}$ with $k,c,r\geq1$ and $T[n]=O(1)$ if $n\leq2$?
1
vote
2answers
101 views

A solution for $f(ax+b)=f(x)+1$

Let $a,b$ be two constant real numbers with $a\neq 0$. Can anyone give a special solution of the functional equation $f(ax+b)=f(x)+1$, where $f:\mathbb{R}\rightarrow \mathbb{R}$? Note. It is a type ...
4
votes
3answers
94 views

Find all functions $f\colon \Bbb R\to \Bbb R$ such that $f(1-f(x)) = x$ for all $x \in \Bbb R$ [duplicate]

Find all functions $f\colon \Bbb R\to \Bbb R$ such that $f(1-f(x)) = x$ for all $x \in \Bbb R$. This is a question from the national olympiad in Germany 2018. All i could do is to try with some ...
5
votes
2answers
111 views

Find all the functions in $\mathbb{R}$, satisfying the given equation $(x+y)(f(x)-f(y)) = (x-y)f(x+y)$ for all $x,y$ in $\mathbb{R}$.

Find all the functions in $\mathbb{R}$, satisfying the given equation $(x+y)(f(x)-f(y)) = (x-y)f(x+y)$ for all $x,y$ in $\mathbb{R}$. I tried to find something like a pattern that would become a ...
4
votes
0answers
58 views

Does there exist a function $f$ such that $f(x)$ is an integer for only finitely many values of $x$?

Define $f : \mathbb{R} \to \mathbb{R}$ such that $f(x)\leq f(y)$ whenever $x\leq y$ and $f^{2018} (z) \in \mathbb{Z}$ $\forall z \in \mathbb{R}$. Does there exist a function $f$ such that $f(x)$ is an ...
3
votes
1answer
64 views

Find smallest possible degree of $f(x)$ from given conditions

Let $p(x)$ be a polynomial of degree strictly less than 100 and such that it does not have $x^3-x$ as a factor. If $$\frac{d^{100}} {dx^{100}} \left(\frac{p(x)}{x^3-x}\right)=\frac{f(x)} {g(x)}$$ ...
4
votes
1answer
65 views

show that $f(x) =0$ [closed]

$f$ is a continous function in $[0,1]$ $$ f\left(\frac{x}{2}\right)+f\left(\frac{1+x}{2}\right) = 3f(x) \qquad \forall x \in [0,1]$$ Show that $f(x)=0$
0
votes
2answers
29 views

Finding continuous funcions satisfying a given condition

The problem asks to determine all functions defined and continuous over the set of reals that satisfy : $$|f(x)-\sin(x)|=|x|$$ for all reals $x$. It's immediate that for any real $x$, we either have ...
1
vote
0answers
59 views

Non-monotic Function $f(x)$ such that $f\circ f(x) =9x+4$

I have to construct a non-monotonic function (defined on any interval) such that $f\circ f(x)=9x+4$. I tried making a $2$ branch function with $f(x)=3x+1$ and $f(x)=-3x-2$, but I don't know how to ...
0
votes
1answer
29 views

Function$f(x)={2x+3xf(x)\over x-1}$ is one to one and an onto function. what is the codomain of f(x)?

$$f(x)={2x+3xf(x)\over x-1}$$ This function is one to one and an onto function. what is the codomain of $f(x)$? How to solve this function?
0
votes
0answers
29 views

Iterative function $f^i(n)=3n$ where $n\in\mathbb{N} \setminus \{ 2^i \}$ never reaches $2^x$

Iterative Function $f^i(n)=3n$, where $x\in\mathbb{N}$ and $n\in\mathbb{N} \setminus \{ 2^x \}$ as $x\to\infty$. How can we show that $f^i(n)$ as $i\rightarrow \infty$, never reaches $2^i$. The ...
1
vote
1answer
38 views

Finding $f(5)$ if $f(2a-b) = f(a) \cdot f(b)$ for all a and b, where $f(x)≠0$

If $f(2a-b) = f(a) \cdot f(b)$ for all a and b, and the function is never equal to 0, find the value of f(5). As such, what I've already tried is simply eliminating all possibilities of what type of ...
1
vote
3answers
54 views

Finding $f(9999)$ if $f(1) = 2$ and $f(mn) = f(m) \, f(n) - f(m+n) + 1001$ for $m$ or $n$ equal to $1$

Given a function $f$ is defined for integers $m$ and $n$ as given: $$f(mn) = f(m)\,f(n) - f(m+n) + 1001$$ where either $m$ or $n$ is equal to $1$, and $f(1) = 2$. The problem itself is to ...
0
votes
1answer
53 views

Proving the function with this property is bijective [closed]

I do not know how to get $f(x)$, so I can see if it is bijective. $$f\colon \mathbb{R}\to \mathbb{R}$$ $$2f(3-2x)+f(2x-2)=x.$$
2
votes
4answers
77 views

if $f(f(n))+f(n)=2n$ find $f$

Let $f:N^{+}\to N^{+}$ and such $$f(f(n))+f(n)=2n$$ find $f$ I think the answer is $f(n)=n$,We prove this by induction. (at last step I can't induction it) It is true for $n=1$ because ...
0
votes
0answers
27 views

A book on functional equations

I want to learn about functional equations for Olympiad. I am a beginner and have not studied it earlier.I request you all to suggest a book that starts from the very beginning and provides good ...
3
votes
1answer
160 views

A functional equation of a matrix

How would one prove the following theorem? $p(A)$ is a nonzero polynomial of the entries of $A$ and satisfies $p(AB)=p(A)p(B)$, for all square matrices $A$ and $B$ of complex numbers. Prove $p(A)=(\...
2
votes
2answers
86 views

Are there any real-valued functions, besides logarithms, for which $f(x*y) = f(x) + f(y)$?

Is there any real-valued function, $f$, which is not a logarithm, such that $∀ x,y$ in $ℝ$ , $f(x*y) = f(x) + f(y)$? So far, all I can think of is $z$ where $z(x) = 0$ $∀ x$ in $ℝ$ EDIT: Functions ...
3
votes
1answer
101 views

Functions satisfying $f(x+y)+f(x-y)=2f(x)g(y)$

Given $f,g:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f$ is not the zero function, and $\forall x\in\mathbb{R},\; |f(x)|\leq 1$, and $\forall x,y\in\mathbb{R},\; f(x+y)+f(x-y)=2f(x)g(y)$, can we ...
-2
votes
0answers
28 views

Question about a estimative in Hernández's article

On page 1143 of the article by Hernández, Rabello and Henríquez, at https://ac.els-cdn.com/S0022247X0601033X/1-s2.0-S0022247X0601033X-main.pdf?_tid=c4504062-1c37-4ac3-a2a4-6d7041b2f579&acdnat=...
8
votes
1answer
101 views

How to solve equations in the form $af^2(x)+bf(x)+cx=0$?

I have not seen any books about solving the equation of the following form: $af^2(x)+bf(x)+cx=0$ where $a$, $b$, $c$ are constants and $f^2(x)=f(f(x))$. We are going to find an expression of the ...
2
votes
1answer
87 views

If $f(f(x))=-f(x)$,does it necessarily follow that $f(x)=-x$?

Is there any counterexample other than $f(x)=0$? The actual problem states: Find a function $f: \mathbb{R} \to \mathbb{R}$, such that $f(f(x+y))=x-f(y)$. By substituting $x=0$, we get $f(f(y))=-f(y)$....
0
votes
1answer
36 views

Functional equation $f(a)·f(b)=\Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$

I am trying to get all possible solutions of the following functional equation: $$f(a)·f(b)= \Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$$ Where {} mean fractional part function. Solutions only need to be ...
2
votes
3answers
79 views

Find all non-negative integer $n$ that satisfy $f(x+1)+f(x-1)=\sqrt nf(x)$

Find all non-negative integer $n$ that there exists a non-periodic function $f:\Bbb R->\Bbb R\quad f(x+1)+f(x-1)=\sqrt nf(x)\forall x$. My attempt: $f(x+2)+f(x)=\sqrt n\cdot f(x+1)$ $\sqrt n \...
1
vote
2answers
83 views

Functions satisfying $f(x)+f(\frac{1}{1-x})=x$ with $x\in\mathbb{R}\setminus\{0,1\}.$ [duplicate]

I have used this identity: if $g(x)=1/(1-x),$ then $$g^{-1}(x)=1-\frac{1}{x},$$ to get all functions satisfying: $f(x)+f(\frac{1}{1-x})=x$ with $x\in\mathbb{R}\setminus\{0,1\},$ but I didn't get a ...
0
votes
1answer
67 views

determine every function $f$ defined for positive numbers

determine every function $f$ defined for positive numbers , having positive values, such that : $f(xf(y))f(y)=f(x+y)$ $f(2)=0$ $f(x)\neq 0$ for every $0\le x<2$ Iproved that $f(x)=0$ for every $...
0
votes
1answer
65 views

Functional equation with particular structure

I have ran into a functional equation and am not sure how to proceed with solving it. I was hoping that it may be a known equation, as it has a particular structure to it: $$ f(x) = a(x) f(x-t) + b(x) ...
3
votes
1answer
79 views

Are constant and identity only analytic functions such that $f(f(x))=f(x)$?

If a function over reals is given by $$f(x) = \sum_{n=0}^\infty a_n x^n$$ and satisfies $$f(f(x))=f(x),$$ does this imply that $f(x) = x$ and $f(x) = c$ are only valid choices for $f(x)$? It seems ...
2
votes
1answer
48 views

Maximisation of a functional under integral constraint

I need your help. I have got a maximisation problem that seems to be "easy" at first sight, and I have also got an intuitive solution for the problem, but I am not able to translate my intuitions into ...
0
votes
0answers
54 views

Differential functional equation [duplicate]

I'm struggling to solve this problem: Let $f$ be a continuous function defined on $[0,+\infty[$, derivable in every point such that $$ f'(x)=2f(2x)-f(x), \ \ \ \ \ \ \forall x>0, $$ and $$ M_n=\...
0
votes
0answers
50 views

Are there any methods for finding analytic solutions to this equation $(xy + \frac1e)e^{F(x,y)} = F(g(y)x , f(y))$

I'm trying to solve the equation $$ \left(xy + \frac1e\right)e^{F(x,y)} = F(g(y)x , f(y)) $$ for $F(x,y)$ where $f$ and $g$ are known functions, analytic in a neighborhood of $0$, with $g(0) = 1$ , $f(...
8
votes
2answers
149 views

Functional equation related to $\sin$: $f(x+y)=f(x)f'(y)+f'(x)f(y)$

Find all differentiable $f:\mathbb R \to \mathbb R$ such that $$\forall (x,y)\in \mathbb R^2, f(x+y)=f(x)f'(y)+f'(x)f(y)$$ It's easy to check that the only constant solution is $0$ and the only ...
5
votes
0answers
112 views

Is this $f(x) = x+1$ the only solution to this functional equation.

I am considering the problem of finding all functions $f:(0,\infty)\rightarrow\mathbb{R}$ satisfying the functional equation: $$f\big(xf(y)+f(x)\big) = 2f(x)+xy.$$ I have been able to prove the ...
0
votes
2answers
75 views

Solving the functional equation $f(1-x)+f(x)=1$

Let $f(1-x)+f(x)=1$ for a function $f:\mathbb{R}\to\mathbb{R}$. What would the be the value of a possible $f(x)$? (I just need an example).
2
votes
1answer
40 views

Does $f_0=x,f_{n+1}(x)=e^{1/f_n(x)}$ converge to $1/W(1)$?

Question: Define $f_0=x,f_{n+1}(x)=e^{1/f_n(x)}$ and $x\neq0$. What does this sequence of functions converge to as $n\to+\infty$? Is it true that $f_n(x)$ converges to the constant $\frac1{W(1)}$, ...
4
votes
1answer
75 views

A functional equation from a regional math olympiad (Dhaka regional, 2017)

Question: For any rational numbers $x$ and $y$, $f(x)$ is a real number and: $$f(x+y) = f(x)f(y) - f(xy) +1$$ Again, $f(2017) \neq f(2018)$ and: $$f\left(\frac{2017}{2018}\right) = \frac{a}{b}$$ where ...
3
votes
2answers
88 views

Find all functions if $f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$ for all $x,y\in\mathbb{R}$.

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$ for all $x,y\in\mathbb{R}$. If we put $x=0$ ...