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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7
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2answers
117 views

Function such that $f(x) f(\pi/2 - x) = 1$

I'm looking for functions that are smooth ($C^\infty$) between $0 < x < \pi/2$ that satisfy the equation $$f(x)\, f(\pi/2-x) = 1$$ on the inteverval $0<x<\pi/2$. I know that the constant ...
7
votes
3answers
175 views

Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$

I got this problem: Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$. (Hint: the solution involves limits at ...
7
votes
1answer
347 views

$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) )$

Suppose $f:\mathbb R\to\mathbb R$ is a strictly decreasing function which satisfies the relation $$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) ) , \quad \forall x , y \in\mathbb R $$ ...
7
votes
3answers
266 views

Does there exist a function such that $f(a)f(b)=f(a^2b^2)?$

Given $S=\{2,3,4,5,6,7,\cdots,n,\cdots,\} = \Bbb N_{>1}$, prove whether there exists a function $f:S\to S$, such that for any positive $a,b$: $$f(a)f(b)=f(a^2b^2),a\neq b?$$ This is 2015 ...
7
votes
2answers
315 views

Defining sine and cosine

We know the following are true about sine and cosine (and that they can be proven geometrically): $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ ...
7
votes
1answer
118 views

All functions $g:\mathbb{N}\to\mathbb{Z}$ such that $m+n~|~g(m)+g(n)$

I just thought about the following question: Find all functions $g:\mathbb{N}\to\mathbb{Z}$ such that $m+n~|~g(m)+g(n)$ for all $m,n\in\mathbb{N}$. Clearly every polynomial $g(X)\in\mathbb{Z}[X]$ in ...
7
votes
2answers
145 views

Continuous functions that satisfies $f(x) + f(1-x) + f(\sqrt{x^2+(1-x)}) = 0$ and $f(\frac12)=0$

$f:[0,1]\rightarrow \mathbb{R}$ is a continuous function which satisfies $$ f(x) + f(1-x) + f\left(\sqrt{x^2+(1-x)}\right) = 0 \text{ and } f\left(\tfrac12\right)=0. $$ Can someone give explicit ...
7
votes
1answer
479 views

How do you solve $f'(x) = f(f(x))$?

A friend told me to solve the following differential equation: $$f'(x)=f(f(x))$$ I have no idea how to solve this! This doesn't seem to be an ordinary differential equation and I can't even solve ...
7
votes
1answer
1k views

Solution for exponential function's functional equation by using a definition of derivative

let $f(0)=1$ and $f'(0)=1$. and $f(x+y)=f(x)f(y)$ for $x,y\in R$. How can I found $f(x)$ by using a definition of derivative?
7
votes
2answers
92 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 ...
7
votes
1answer
187 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
7
votes
1answer
144 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
7
votes
1answer
133 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 ...
7
votes
3answers
131 views

Functions proof.

Find all functions $$f: \mathbb{Z} \rightarrow \mathbb{Z}$$ such that $$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$$ for all integers $$a, b, c$$ satisfying $$a+b+c=0$$ I have no idea how to ...
7
votes
0answers
98 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
7
votes
0answers
134 views

How to prove subadditive function?

Let $f: [0, \infty) \to \Bbb R$ be a function satisfying the following conditions: (1) For any $x,y \geq 0, f(x+y) \geq f(x) + f(y)$. (2) For any $x \in [0,2], f(x) \geq x^2 - x$. Prove that, for ...
7
votes
0answers
111 views

Any chance for a solution of $f[f(x)]=x^2+1$? [duplicate]

Not sure if this is a closed or open question. But the question is suppose $f[f(x)]=x^2+1$ then what is $f(x)$? Though the question does not refer to the domain let us suppose it is defined on $\Bbb ...
7
votes
0answers
65 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 ...
6
votes
3answers
883 views

Classifying Functions of the form $f(x+y)=f(x)f(y)$ [duplicate]

Possible Duplicate: Is there a name for such kind of function? The question is: is there a nice characterization of all nonnegative functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that ...
6
votes
3answers
362 views

Solve $f(f(n))=n!$

What am I doing wrong here: ( n!=factorial ) Find $f(n)$ such that $f(f(n))=n!$ $$f(f(f(n)))=f(n)!=f(n!).$$ So $f(n)=n!$ is a solution, but it does not satisfy the original equation except for ...
6
votes
2answers
434 views

Find all $f(x)$ if $f(1-x)=f(x)+1-2x$?

To find one solution I assumed that $f$ is even and rewrote this as $f(x-1)-f(x)+2x=1.$ By just thinking about a solution, I was able to conclude that $f(x)=x^2$ is a solution. However, I am sure that ...
6
votes
4answers
329 views

3rd iterate of a continuous function equals identity function

If $ f: \mathbb{R} \to \mathbb{R} $ is continuous, and $\forall x \in \mathbb{R} :\;(f \circ f \circ f)(x) = x $, show that $ f(x) = x $. The condition that $f$ is continuous on $\mathbb{R}$ is ...
6
votes
2answers
703 views

Functional Equation $f(x+y)=f(x)+f(y)+f(x)f(y)$

I need to find all the continuous functions from $\mathbb R\rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)+f(x)f(y)$. I know, what I assume to be, the general way to attempt these problems, but I ...
6
votes
1answer
298 views

All functions $\frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + \frac{2x^2}{f(y)}\right)$

How can I find all continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}^+$ such that $$\frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + ...
6
votes
2answers
129 views

Show that $f(x) = x$ if $f(f(f(x))) = x$.

If $f: \mathbb{R} \to \mathbb{R}$ and $f$ is strictly increasing, show that $f(x) = x$ if $f(f(f(x))) = x$. So this compulsorily ESTABLISHES that $f(x) = x$ only, and no other solution. So, merely ...
6
votes
3answers
168 views

functions satisfying $f(x)=2f(2x)$

How can I prove every smooth real function f satisfying $f(x)=2f(2x)$ is of the from $f(x)=k/x$ where $k$ is a constant? I have tried this for integers that can be divided by $2^n$, but then I cannot ...
6
votes
2answers
164 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.
6
votes
2answers
128 views

Problem about functional equation (Bulgarian selection team test)

This problem was taken from the bulgarian selection team test for the 47th IMO and appeared in a chinese magazine, I came across it in my own training. http://www.math.ust.hk/excalibur/v10_n4.pdf ...
6
votes
4answers
586 views

Find a solution for $f\left(\frac{1}{x}\right)+f(x+1)=x$

Title says all. If $f$ is an analytic function on the real line, and $f\left(\dfrac{1}{x}\right)+f(x+1)=x$, what, if any, is a possible solution for $f(x)$? Additionally, what are any solutions for ...
6
votes
2answers
128 views

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ . . .

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that $$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$ I've tried subbing in heaps of values but I keep getting things like ...
6
votes
3answers
286 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 ...
6
votes
2answers
104 views

Solution of Equation $ 1 + x + \frac{1\cdot3}{2!}x^2 + \frac{1\cdot3\cdot5}{3!}x^3 + \ldots = \sqrt{2} $

I have an equation whose left side is a infinite series . I can solve the equation if I am able to find a close form of the series . The equation is as follows : $$ 1 + x + \frac{1\cdot3}{2!}x^2 + ...
6
votes
2answers
214 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 ...
6
votes
2answers
112 views

Functional Equation f(x) = f(x/2)

Find all functions $f$ satisfying the property that $$ f(x) = f(x/2) $$ for all $x \in \mathbb{R}$ So far I've come up with the following assumptions: -$f$ is periodic, i.e of form $f(x) = A ...
6
votes
2answers
519 views

Uniqueness of Tetration

Let $f(0)=1$ and $f(x+1)=2^{f(x)}$ Also let f be infinitely differentiable. Then does f exist and is it unique? If f is merely continuous, then any continuous function such that f(0)=1 f(1)=2 ...
6
votes
3answers
198 views

Is there a non-constant function $f:\mathbb{R}^2 \to \mathbb{Z}/2\mathbb{Z}$ that sums to 0 on corners of squares?

A problem in the 2009 Putnam asks about functions $f:\mathbb{R}^2 \to \mathbb{R}$ such that whenever $A,B,C,D$ are corners of some square we have $f(A)+f(B)+f(C)+f(D)=0$. Without spoiling the problem ...
6
votes
1answer
258 views

$ f(x+f(x+y))=f(x-y)+f(x)^2 \quad \forall x,y\in \mathbb R$

We have to find all functions $f\colon \mathbb R\to\mathbb R$ such that f: $$\forall x,y\in \mathbb R \quad f(x+f(x+y))=f(x-y)+f(x)^2.$$ Could somebody help me solve this problem? Thank you.
6
votes
3answers
282 views

Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$

I recently came across this equation : $$\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$$where $f \in \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. I've done the following, but I'm stuck at ...
6
votes
4answers
79 views

Find all polynomials with real coefficients that satisfy $(x^2-6x+8)P(x)=(x^2+2x)P(x-2)$

Find all polynomials with real coefficients that satisfy $$(x^2-6x+8)P(x)=(x^2+2x)P(x-2)\forall x\in\Bbb R$$ My work; $$\frac{P(x)}{P(x-2)}=-\frac{4}{x-2}+\frac{12}{x-4}+1\tag{1}$$ ...
6
votes
1answer
99 views

Functional Equation $f(mn)=f(m)f(n)$.

If $f: \mathbb N \mapsto \mathbb N$ is one-to-one and $f(mn) = f(m)f(n)$, what is the smallest possible value of $f(999)$? Easily $f(1)=1$, and I think $f(n)=n$ must be the only map, but not able to ...
6
votes
1answer
285 views

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfied that: $f(xy)+f(x+y)=f(xy+x)+f(y)\quad\forall x,y \in \mathbb{R}$

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfied that: $$f(xy)+f(x+y)=f(xy+x)+f(y)$$ $\forall x,y \in \mathbb{R}$ I have tried that : $P(y;x)-P(x;y)$: ...
6
votes
1answer
897 views

$f(m + f(n)) = f(f(m)) + f(n)$

I found this one in the list of IMO'96 (3) problems and decided to have a go at it, but could not complete the solution. So $m$ and $n$ are non-negative integers and $f$ takes values in the same set: ...
6
votes
1answer
248 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 ...
6
votes
2answers
203 views

Find $f(x)$ satisfy $f(2x)=2f(x)+x$

I would appreciate if somebody could help me with the following problem: Find $f(x)$, given that: $f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous at $x=0$, and ...
6
votes
1answer
179 views

Functional Equation - Am I right?

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$ So here's my solution, If $x=y=0$, $2f(0)=2f(0)^2$ $\implies f(0)=0$ or $f(0)=1$. Case $1$: ...
6
votes
2answers
271 views

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$. Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
6
votes
2answers
327 views

Proving a function is constant when $f(x)f(y) + f(\frac{a}{x})f(\frac{a}{y}) = 2f(xy)$

I've been working on the following homework problem: Consider a function $f : (0,∞) → \mathbb{R}$ and a real number $a > 0$ such that $f(a) = 1$. Prove that if $f(x)f(y) + ...
6
votes
1answer
106 views

Can we give a definition of the cotangent based on a functional equation?

I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...
6
votes
1answer
324 views

Iterated polynomial problem

Polynomial $P$ satisfies $P(n)>n$ for all positive integers $n$. Every positive integer $m$ is a factor of some number of the form $P(1),P(P(1)),P(P(P(1))),\ldots $. Prove that $P(x)=x+1$.
6
votes
2answers
196 views

Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$ 1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}} $$ EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a ...