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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11
votes
4answers
389 views

$f(16x)=16f(x) $ and $ f$ is continuous

$f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(16x)=16f(x)$ for every real $x$. Should it be $f(x)=ax$? How can I prove that?
2
votes
1answer
107 views

functional equation problem in competition

Find all $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that $f(1)=2$ and $f(xy)=f(x)f(y)−f(x+y)+1$. for all $x,y \in \mathbb{Q}$. thank you very much!
7
votes
3answers
308 views

$f(x^2) = 2f(x)$ and $f(x)$ continuous

I ran into a problem recently where I obtained the following constraint on a function. $$f(x^2) = 2f(x) \,\,\,\, \forall x \geq 0$$ and the function $f(x)$ is continuous. Can we conclude that $f(x)$ ...
0
votes
2answers
38 views

Implicit equation - $G(x,y)=0$

I'm confused about some points in implicit equation ... From my recitation class - $G(x,y)=0$ provides - $y=f(x)$ . And $f'(x)=\frac{dy}{dx}$ and about $G'(x,y)=0$ we use - ...
3
votes
2answers
88 views

Solution to functional equation

I have the following functional equation: $$f(a+b)=f^{n-1}(a)f(b)+f^{n-2}(a)f^{1}(b)+...+f(a)f^{n-1}(b)=\sum_{k=0}^{n-1}f^{n-1-k}(a)f^{k}(b)$$ where $a,b$ are complex and the function $f$ is an ...
0
votes
2answers
543 views

Find all functions that satisfy $f(f(x)+y)=2x+f(f(y)-x)$

Find all the function satisfy: $$f(f(x)+y)=2x+f(f(y)-x), \forall x , y \in \mathbb{R}$$ I have tried that: Let $x:=-x $ we have : $$f(f(y)+x)=2x+f(f(-x)+y) ,(1) $$ Then in $(1)$ $x:=y;y:=x$ we have ...
4
votes
3answers
99 views

An odd function satisfying $g(1-t)+g(1+t)=-t$

I am looking for a continuously diffferentiable odd function $g$ such that $$g(1+t)+g(1-t)=-t$$ for all $t\in\mathbb{R}$. Is this possible?
5
votes
1answer
152 views

Generalization of Cauchy's functional equation

We know that if $f(x+y)=f(x)+f(y)$ and $f$ meets some "reasonable" conditions, then $f$ is linear. I've been considering the following extension: consider the reals under some unknown group operation ...
1
vote
1answer
91 views

“Foldable” functions

Suppose $f:2^X\to X$ satisfies $f(x_1,\dots)=f(f(x_1,x_2),x_3,\dots)$. Min, max and sum are three such examples. I've been calling these functions "foldable" because they bear some similarity to ...
0
votes
1answer
214 views

More general reflection formula for Gamma function

It is know that $\Gamma(z)\, \Gamma{(1-z)}=\pi \csc( \pi z)$. Is there any formula for $\Gamma{(a+z)}\Gamma{(a-z)}$ where $a$ is a rational number, i. e., $a=p/k$ with $p, k$ integers and $z$ is ...
1
vote
1answer
100 views

functional equations

Find all solutions $f:\Bbb R^2 \to \Bbb R$ satisfying $$ f(xu-yv, yu+xv)=f(x, y)f(u, v). $$ Solution of the following equation $$ f(xu+yv, yu-xv)=f(x, y)f(u, v) $$ is known as $$ f(x,y)=m(x^2+y^2), ...
5
votes
1answer
143 views

Does a function exist with the property $f(-n^2+3n+1)=(f(n))^2+1$?

Let $f:\mathbb{R}\to \mathbb{R}$ be a function which fulfills for every $n \in \mathbb{N}$ $$f(-n^2+3n+1)=(f(n))^2+1$$ Is it possible that such a function exists?
14
votes
4answers
602 views

Polynomial: $p(x) = p(x+3)$.

Determine polynomial $p(x)$ s.t. $p(x) = p(x+3)$. Just by looking at the above equation, it immediately appears that p has got to be some kind of constant function. I thought it might also be a ...
4
votes
2answers
90 views

Functional equation $m(x^y)=m(x)+m(y)$.

Find all functions $m : \mathbb{R}^+ \to \mathbb{R}^+$ such that$$m(x^y)=m(x)+m(y)$$
1
vote
1answer
99 views

Properties of $g$ satisfying $f(x,x)[\nabla_{x}^{2}g(x,y)]_{x=y}+2[\nabla_{x}f(x,y)]_{x=y}\cdot[\nabla_{x}g(x,y)]_{x=y}=0$ for all $f$

Suppose that $g:\mathbb{R}^{2n}\rightarrow\mathbb{R}$ is a function such that: ...
6
votes
1answer
83 views

How can find this function by $x\in \mathbb{Q}^+$

Let $f:\mathbb{Q}^+\to \mathbb{Q}^+,f(x)+f(1/x)=1 $ and $f(2x)=2f(f(x)),x\in \mathbb{Q}^+$, prove that $$f(x)=\dfrac{x}{x+1},x\in \mathbb{Q}^+$$ This Problem from my student.
12
votes
1answer
197 views

How can prove this equation.

if $a+b=c+d=e+f=\dfrac{\pi}{3}$, $\dfrac{\sin{a}}{\sin{b}}\cdot\dfrac{\sin{c}}{\sin{d}}\cdot\dfrac{\sin{e}}{\sin{f}}=1$, Prove that: ...
11
votes
3answers
295 views

Functions satisfying $\sum_{n=0}^k(-1)^n\binom{k}{k-n}f^{k-n}(x)=0$.

This question was motivated by this one. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. Define $f^n(x)=f\circ f\circ\cdot\cdot\cdot\circ f$, $n$ times, $f^0(x)=x$ and $k\geq 2$ a ...
19
votes
1answer
555 views

Functions $f$ satisfying $ f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R}$.

How to prove that the continuous functions $f$ on $\mathbb{R}$ satisfying $$f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R},$$ are given by $$f(x)=x+a,a\in\mathbb{R}.$$ Any hints are welcome. Thanks.
5
votes
1answer
295 views

$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 \right ),\forall x\in \mathbb{R},m\in \mathbb{Z^+}$

Find all the function $f:\mathbb{R}\rightarrow \mathbb{R}$ sastisfied that $f$ continuous on $\mathbb{R}$ and $$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 ...
6
votes
2answers
167 views

Map that satisfies $f(\lambda x) = \lambda f(x)$ but not $f(x+y) = f(x)+f(y)$

Could you give me example of maps $f:\mathbb R \to \mathbb R$ that satisfy $$ f(\lambda x) = \lambda f(x) \quad \forall x,\lambda \in \mathbb R $$ but not $ f(x+y) = f(x)+f(y) $? Thanks in advance.
4
votes
2answers
228 views

The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$

I came across the following problem that says: The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$ is which of the following? $(1)0\space (2)1 ...
0
votes
2answers
57 views

$f(\vec{x_1}+\vec{x_2})=f(\vec{x_1})f(\vec{x_2})$

What is the general solution to $f(\vec{x_1}+\vec{x_2})=f(\vec{x_1})f(\vec{x_2})$ where $\vec{x}$'s are in discrete vector space $x\in \{n_1\vec{e_1}+n_2\vec{e_2}+n_3\vec{e_3},n_1,n_2,n_3 \in Z\}$?
3
votes
4answers
284 views

Sufficient condition for a function to be affine

If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form ...
0
votes
0answers
96 views

Interpretation of functional equation of dedekind eta function

It's well known that the Dedekind eta function defined by $\eta(z) = \displaystyle e^{\frac{\pi i z}{12}} \prod_{n=1}^{\infty} (1 - e^{2 \pi i n z}) $ converges for $z$ in the upper half plane to a ...
0
votes
2answers
125 views

$f(xy)=f(x(1-y)) \forall x,y\in (0,1)$, Find all $f:(0,1)\rightarrow \mathbb{R}$

Find all functions $f:(0,1)\rightarrow \mathbb{R}$ such that $f(xy)=f(x(1-y))$ for all $x,y\in (0,1)$.
1
vote
2answers
63 views

$\frac{f(x_1)}{f(x_2)} = \log(\frac{x_1}{x_2}) \implies f(x)=\;?$

If $$\frac{f(x_1)}{f(x_2)} = \log\left(\frac{x_1}{x_2}\right),$$ what is $f(x)$? I mean the simplest form of $f(x)$, and what math technique you use to solve this problem? Thanks.
6
votes
2answers
243 views

functional equations for trigonometric functions

It is well known that the following system of functional equations: $\begin{cases} f(x+y)=f(x)f(y)-g(x)g(y) \\ g(x+y)=f(x)g(y)+g(x)f(y) \end{cases}$ admit the solution $(f,g)=(\cos,\sin)$. Are there ...
10
votes
1answer
196 views

Solution(s) to $f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y))$

I would like to know the solutions of the functional equation: $$f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y)), \forall x,y\in\mathbb{R}$$ where $f:\mathbb{R}\rightarrow\mathbb{R}$. I have already determined ...
2
votes
1answer
89 views

How to solve the functional equation : $T(n)=(\log n)T(\log n)+n$

I want to solve the following functional equation using any ways: $$T(n)=(\log n)T(\log n)+n$$
7
votes
0answers
72 views

What are the densities of branches of the euclidean tree?

The Euclidean algorithm shows how all coprime pairs of positive integers can be uniquely obtained from the pair $(1,1)$ by applying the two operations $(a,b) \to (a+b,b)$ and $(a,b) \to (a,a+b)$. (or ...
11
votes
1answer
475 views

Find all the function that satisfy : $f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$

Find all the function that satisfy : $$f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$$ I only find $f(0)=0$ but I can't prove $f(x)=2x$
106
votes
8answers
5k views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?

I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success: Find a continuous ...
1
vote
1answer
43 views

Funcional Equations:I'm confused [duplicate]

I need help with this: "Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that: $$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$ Or : ...
0
votes
2answers
79 views

What's the solution of the functional equation

I need help with this: "Find all functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, with $g$ injective and such that: $$f(g(x)+y) = g(f(x)+y), \mbox{ for all } x, y \in \mathbb{Z}.$$
7
votes
4answers
787 views

If $f(x/2)=f(x)/2$, then $f(x)=f'(0)x$

Let $f:\mathbb R \to \mathbb R$ be differentiable such that $f(x/2)=f(x)/2$ for any $x\in \mathbb R$. How can I prove that $f(x)=f'(0)x$, for any $x\in \mathbb R$? It seems easy, but I don't know why, ...
0
votes
2answers
150 views

How to create equations to measure time spending in executing algorithms?

I made a program with two functions to calculate factorial. The first uses loops to made de calculations, and the second uses recursive calls to get the same result. The same program measures the ...
1
vote
1answer
102 views

Are all functions on vectors in $GF(2^n)$ representable as polynomial functions?

In $GF(2)$, any function from an n-dimensional vector to a number is equal to a polynomial function of n variables. (See proof below.) Question: Is this true for other $GF$, especially $GF(2^8)$? ...
1
vote
1answer
96 views

simplifying “$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}$”

Is this equality correct? For finite sets $A$ and $B_a$ (where $a\in A$), we have: $$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}=\sum_{f\in \prod_{a\in A}B_a}\quad \prod_{a\in A}{h(a,f(a))}$$
1
vote
1answer
79 views

What's the solution of the functional equation?

I need help with this: "Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that: $$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$
6
votes
2answers
4k views

How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?

This is part of a homework assignment for a real analysis course taught out of "Baby Rudin." Just looking for a push in the right direction, not a full-blown solution. We are to suppose that ...
7
votes
3answers
936 views

Find all the functions which satisfy a given functional equation

I was given by a friend of mine the following problem. Find all the functions $f:\mathbb R\to\mathbb R$ which satisfy the following functional equation $$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.$$ We ...
9
votes
2answers
89 views

Uniqueness of solution for a functional equation

Let $h \in \mathcal{C}:=C([-a,a])$, where $a>0$. Prove that there exists a unique function $f \in \mathcal{C}$ such that $$ f(x)=\frac{x}{2}f\Big(\frac{x}{2}\Big)+h(x)\quad \forall x \in [-a,a]. $$ ...
5
votes
1answer
296 views

Strictly monotone functions

What are the strictly monotone functions $f\colon (0,\infty)\to (0,\infty)$ which satisfy $x= f(\tfrac{x^2}{f(x)})$ for $x>0$. I cannot find any other than $f(x)=x$.
1
vote
1answer
35 views

Plotting for solution for $y=x^2$ and $x^2 + y^2 = a $

Consider the system $$y=x^2$$ and $$x^2 + y^2 = a $$for $x>0$, $y>0$, $a>0$. Solving for equations give me $y+y^2 = a$, and ultimately $$y = \frac {-1 + \sqrt {4a+1}} {2} $$ (rejected ...
2
votes
2answers
738 views

Attractive fixed-point?

I'm sorry in advance for how specific this problem is. I've been trying to make it as generic as possible but this is as far as I could get. I want to prove that a fixed point is attractive. I didn't ...
2
votes
1answer
87 views

Connecting functional identity of a function with its image set

Okay, now I really need real help, maybe the task is not too heavy but I do not know the easy way to solve this. Let´s start with the problem, now. Suppose that we have some function $f: \mathbb N ...
2
votes
2answers
191 views

How to find $f(x)+f(x+1) = e^x$?

Let $x,a,b$ be real numbers and $f(x)$ a (nongiven) real-analytic function. How to find $f(x)$ such that for all $x$ we have $f(x)+af(x+1)=b^x$ ? In particular I wonder most about the case $a=1$ ...
5
votes
1answer
169 views

Question about a functional equation

We are looking at a theorem which characterizes the affine term structure (ats) models in interes rate theory. What follows is from "Filipović, D. (2009): "Term-structure models: A graduate course", ...
3
votes
2answers
131 views

Average of function, function of average

I'm trying to find all functions $f : \mathbb{R} \to \mathbb{R}$ such that, for all $n > 1$ and all $x_1, x_2, \cdots, x_n \in \mathbb{R}$: $$\frac{1}{n} \sum_{t = 1}^n f(x_t) = f \left ( ...