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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6
votes
3answers
616 views

I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$

I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)$. I know that there are other questions that are asking the same thing, but I'm trying to figure this out ...
2
votes
2answers
148 views

Easy functional equation

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that: $$f(2f(x)+f(y))=2x+f(y)\qquad \forall x,y \in \mathbb{R}.$$ If you put $x=y=0$, you get $f(3f(0))=f(0)$. What deductions about ...
0
votes
1answer
63 views

Help creating “Halving” Equation

I am not sure if this is the correct site to ask this but here goes, I have a pseudocode for a program algorithm that I am trying to turn into an equation. Say I have a variable X. Now if X is 8, ...
3
votes
1answer
180 views

Cauchy's functional equation - a generalisation? (do additive maps have to be continuous?)

If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be ...
2
votes
1answer
93 views

Functional equation on $\mathbb Q_p^\star$

I am trying to find solutions of functional equation $f(xy)=f(x)+f(y)$ for all $x,y\in\mathbb Q_p\setminus\{0\}=\mathbb Q_p^\star$. Where $f:\mathbb Q_p^\star\to\mathbb R$. I know some solutions: 1) ...
11
votes
4answers
380 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
95 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!
-1
votes
2answers
68 views

Context problems of Number theory and functional equation

I can't solve the following problems, please help. 1) Find all primes $p$ and $q$ such that $p^q+q^p$ is a prime. 2) Solve $2^x+3^y=z^2$ in integers. 3) Find all $f: \mathbb{Q} \rightarrow ...
7
votes
3answers
283 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
37 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
84 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
407 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
96 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
129 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
82 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
131 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
77 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
141 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
558 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
89 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
95 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
82 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.
11
votes
1answer
191 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
281 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 ...
18
votes
1answer
481 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
236 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
163 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
195 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
55 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
174 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
76 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
116 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.
5
votes
2answers
191 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 ...
7
votes
0answers
154 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
0answers
67 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
63 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 ...
10
votes
1answer
411 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 $f(x)=2x$
94
votes
7answers
4k 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
64 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
687 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
140 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
99 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 $GF(2^8)$? ...
5
votes
1answer
113 views

Maximum cubic function

Let $f(x)=x^{3}+ax^{2}+bx+c$ with a, b, c real. Show that $$\frac{1}4 \le \max_{-1 \le x \le 1\hspace{2mm}} |f(x)|=M$$ and find all cases where equality occurs.
0
votes
1answer
73 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
66 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: $$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$
4
votes
2answers
1k 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
565 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
83 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 $$ f(x)=\frac{x}{2}f\Big(\frac{x}{2}\Big)+h(x)\quad \forall x \in [-a,a]. $$ ...