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 (functions) or (elementary-set-theory).

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0
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3answers
102 views

Can there be a function $f\colon\mathbb Q_{+}^{*}\to \mathbb Q_{+}^{*}$ such that $f(xf(y))=\frac{f(f(x))}{y}$?

Problem: Can an $f$ function be created where:$$f\colon\mathbb Q_{+}^{*}\to \mathbb Q_{+}^{*}$$ The function is defined on the set of fully positive rational numbers and is achieved: $\forall(x,y)\in \...
1
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1answer
38 views

Find the function for two different condition [closed]

Find all function $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation: If $f(x+y) \leq f(x)f(y); f(xy)=f(x)f(y)$ Otherwise, $f(xy)=f(x+y)$
2
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2answers
49 views

Finding a variable $n$ that satisfies the functional composition in which $f(f(f(f(n))))=3.$

Let $f(x) = x^2-2x$. How many distinct real numbers $c$ satisfy $f(f(f(f(n)))) = 3$? I have not found any good way to do this problem. I have just resorted to start with brute force: We start with $$...
0
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4answers
97 views

Spivak's Calculus Chapter 3 Problem 25 [duplicate]

Find a function f(x) such that g(f(x)) = x for some g(x), but such that there is no function h(x) with f(h(x)) = x. I think I've got something backwards in my head because I can easily find the ...
1
vote
1answer
23 views

Can we find $f$ such that $|(x+x^{-1}) - (a+a^{-1})| < f(|x - a|)$

Proposition. Assume $x$ and $a$ are elements of $\mathbb{R}_{\neq 0}$. Then if $x$ is near to $a$, we can deduce that $x^{-1}$ is near to $(a+a^{-1}) - x.$ Proof. Since the function $x^{-1} - (a+a^{-...
1
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2answers
98 views

All solutions of $f(w+z)=f(w)f(z)$, $f(1)=e$

Let $w=u+iv$ and $z=x+iy$ with $u,v,x,y\in\mathbb{R}$. Is the function $$f:\, \mathbb{C}\to\mathbb{C},\, f(z)=e^x\cos y+ie^x\sin y$$ the only solution of the functional equation $$f:\, \mathbb{C}\to\...
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0answers
28 views

A general monotonic rescaling function

I wanted to find a general monotonic rescaling function (don't know if the naming is right) that can rescale a list of values according to 2 parameters : alpha in [...
3
votes
1answer
68 views

Possible solutions to implicit functions

Which $z(x,y)$ are possible solutions from the following equations? \begin{align} S(z) &= xP(y)+Q(y),\\ T(z) &= y M(x)+N(x),\\ G(z) &= y+x F(z) \end{align} Where the functions $S,P,Q,T,M,N,...
4
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0answers
58 views

A functional equation in two complex variables

Let $X$ be a compact metric space, or just $X=\mathbb T$, the unit circle, if it helps. We consider only continuous, complex-valued functions on $X$. Let $\varepsilon >0$. Is there $\delta > ...
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0answers
26 views

Good online references for solving functional equations? [duplicate]

I was wondering what are the main tricks one can try when dealing with a functional equation, like trying values, checking injectivity, surjectivity, bijectivity... In particular, how is one sure ...
3
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2answers
82 views

Functions satisfying $f(x)f(y)=2f(x+yf(x))$ over the positive reals

From the IMO shortlist: We denote by $\mathbb{R}^+$ the set of all positive real numbers. Find all functions $f: \mathbb R^+\rightarrow\mathbb R^+$ which have the property: $$f(x)f(y)=2f(x+yf(x))$$ ...
21
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1answer
496 views

Solve the functional equation $f(x)f(1/x)=f(x+1/x)+1$, $f(1)=2$, where $f(x)$ is a polynomial.

Solve the functional equation $$f(x)f(1/x)=f(x+1/x)+1,\ f(1)=2,$$ where $f(x)$ is a polynomial. It is easy to check that $f(x)=x+1$ is a solution. Are there any other solutions? My attempt is ...
3
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2answers
91 views

How can I determinine all functions or a function f?

Determine all functions $f$ in $f(x+1)=2f(x)$, for all $x$ in real number. So I let $x$ be $x+1$. Then I have $f((x+1)+1)=2f(x+1)$. But since there is a new function $f(x+2)$, I couldn't determine the ...
4
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0answers
42 views

$f : \mathbb{R} \to \mathbb{R}$, $f \in \mathscr{C}^{0}$. Find all $f \not \equiv 0$ satisfying a functional equation

This problem is from an AoPS thread post. $\blacksquare~$ Problem: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function that satisfies the functional relation $$ f(x+y)=a^{x y} f(x) f(...
2
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1answer
59 views

Functional equation in $\mathbb{Q}^+$

Find every function $f:\mathbb{Q}^+\to \mathbb{Q}^+$ Such that: $f(x+1)=f(x)+1, \forall x\in \mathbb{Q}^+$ and $f(x^2)=f^2(x), \forall x\in \mathbb{Q}^+$ All i have managed doing is showing that $f(x)=...
1
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1answer
48 views

How does WolframAlpha solve this recursion?

I have the following recursion: $$x_n=\frac{n-1}{n}x_{n-1}+\frac{1}{n}\left\lfloor\frac{n}{2}\right\rfloor.$$ WolframAlpha gives a solution to this recursion as $$x_n=\frac{C_1+\left\lceil\frac{1-n}{2}...
2
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1answer
36 views

Solve a power tower of function compositions

Lets begin with a simple functional equation ;) Find all functions $f: \mathbb{N}\to\mathbb{N}$ st. $f(x)=x+1$ for all $x \in \mathbb{N}$. I know what you are thinking, "math.SE is for ...
5
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0answers
69 views

What is a solution to the recurrence relation $f(n) = f(n-1) +f\Big(\left\lfloor \frac{n}{2} \right\rfloor\Big)$?

Let $\mathbb{N}=\{1,2,3,\ldots\}$. Find a closed form or an asymptotic form of $f: \mathbb{N} \to \mathbb{N}$, where $f$ satisfies $f(1) = 1$ and $$f(n) = f(n-1) + f\bigg(\left\lfloor \frac{n}{2} \...
0
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1answer
50 views

Polynomial in two variables such that $P(x,y)=P(x+y,x-y)$ [closed]

$P:R\times R \to R\times R$ is a polynomial with real coefficients. It is given that $P\left(x,y\right) = P\left(x+y,x-y\right)$. Find all such $P$.
8
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1answer
64 views

Find all functions $f:\Bbb R^+\to\Bbb R^+$ s.t. for all $x\in \Bbb R^+$ the following is valid: $f\bigg(\frac{1}{f(x)}\bigg)=\frac{1}{x}$

Find all functions $f:\Bbb R^+\to\Bbb R^+$ s.t. for all $x\in \Bbb R^+$ the following is valid: $$f\bigg(\frac{1}{f(x)}\bigg)=\frac{1}{x}$$ I tried to substitute $\frac{1}{x}$ for $x$ and compare ...
0
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0answers
19 views

Cauchy Functional Equation using Laplace transform?

I was just curious. is it possible to solve the Cauchy Functional Equation using Laplace transform or other frequency domain transform? $f(t_1+t_2) = f(t_1) + f(t_2) ~~~\Rightarrow~~~f(t)=c\cdot t$
0
votes
2answers
76 views

If $f(2n)=\frac1{f(n)+1}$ and $f(2n+1)=f(n)+1$ for all $n\in\Bbb N$, then find $n$ such that $f(n)=14/5$. [duplicate]

The set $\mathbb{N}$ is the set of nonnegative integers. Let $ f : \Bbb{N} \rightarrow \Bbb{Q}$ be defined such that 1.) $f(2n) = \dfrac{1}{f(n)+1}$ for all integers $n>0$, and 2.) $f(2n + 1 ) = f(...
12
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2answers
276 views

Find all function $f:\mathbb{R}^+\to \mathbb{R}$ if $xf(xf(x)-4)-1=4x$

Find all function $f:\mathbb{R}^+\to \mathbb{R}$ such that for all $x\in\mathbb{R}^+$ the following is valid: $$xf\big(xf(x)-4\big)-1=4x$$ All I could do is: $f(x)> {4\over x}$ for all $x$ so $f(...
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0answers
34 views

Functional Equation in the rationals [duplicate]

Find all the possible functions $f:\mathbb{Q}\to\mathbb{Q}$ such that $f(1)=2$ and $f(xy)=f(x)f(y)-f(x+y)+1$ I managed to find the function for the set of natural number, by putting $x=1$ and $y=n$ ...
1
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2answers
48 views

Function satisfying some constraints

I am in a hunt for a continuous function $Q : [0,1] \to \mathbb R$ that satisfies the following criteria: $Q(0) = 0$ $Q(1) = 1$ $Q'(x) \geq 0$ for all $x \in [0,1]$ $\int_0^1 P(x)Q(x)= 0.7$, where ...
1
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3answers
58 views

Show that the following power series satisfies this functional equation $f\left(\frac{2x}{1+x^2}\right)=(1+x^2)\,f(x)$.

Show that the following power series satisfies this functional equation $$f\left(\dfrac{2x}{1+x^2}\right)=(1+x^2)f(x)\,,$$ where the series given is $$f(x)= 1+\dfrac{1}{3}x^2+\dfrac{1}{5}x^4+\dfrac{1}{...
0
votes
1answer
28 views

Solving simple functional relation

Satisfying the boundary conditions $$ y(0)=1, y(1)=2 $$ What general /particular functions obey $$ 1) \quad y(x) \,y(x+1)= 2,$$ and $$2)\quad \dfrac{y(x)}{y(x+1)}=\dfrac{1}{2}? \;$$
0
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0answers
63 views

Example of $f(x)-f(y)=g(x-y)$

we know that for linear function $f(x)-f(y)=f(x-y)$ for all $x, y\in D$. Now, I am wondering, if we replace the $f$ in the right hand side of the equality by some nonlinear $g$, are there any other ...
1
vote
3answers
129 views

if $f(\frac{x+y}{2}) =\frac{f(x)+f(y)}{2}$ then find $|f(2)|$ [closed]

if the following functional equation $$f\bigg(\frac{x+y}{2}\bigg) =\frac{f(x)+f(y)}{2} \quad \text{ holds for all real }~ x ~\text{ and }~ y$$ If$f'(0)$ exists and equals to $-1$ then find $|f(2)|$....
6
votes
1answer
87 views

Let $f:\mathbb{R}\to(0,\infty)$ be a differentiable function. For all $x\in\mathbb{R}$ $f'(x)=f(f(x)).$ Then show that such function does not exists [duplicate]

What i have done is very small. $$f'(x)=f(f(x))\implies f(f'(x))=f(f(f(x)))$$Now $$f(f(f(x)))=f'(f(x))$$Hence$$f(f'(x))=f'(f(x))$$Now i am blank. What to do for the proof
1
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0answers
102 views

If $f:\mathbb{R}\setminus\{0,1\}\to\mathbb{R}$ satisfies $f(x)+2f\left(\frac 1 x\right)+3f\left(\frac x {x-1}\right)=x$, then $8f(4)=?$

Let $f: \Bbb{R}\setminus\{0,1\}\to \Bbb{R}$ be a function such that $$f(x)+2f\left(\frac 1 x\right)+3f\left(\frac x {x-1}\right)=x\text{ for all }x\neq 0,1\,.$$ Then, $8f(4)=$ ? My Attempt. Is it ...
-1
votes
2answers
98 views

Solving functional equation $f(x)+f(1/x)=f(x)\cdot f(1/x)$

Let a function $f(x)$, and $x$ not equal to zero be such that: $f(x)+f(1/x)=f(x)\cdot f(1/x)$ then $f(x)$ is ? I tried differentiating it but did not find any useful outcome. answer given at back of ...
3
votes
1answer
75 views

Functional equation for $\eta(s)$ following Riemann's $2^{nd}$ method.

Being \begin{equation*} \eta(s)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots \end{equation*} and following Riemann's second ...
7
votes
4answers
369 views

Dirac delta solutions

I am going through some lecture notes on Fourier transforms (here) and it is stated without proof (example 2.16 on page 29) that the general solution to the equation $$x f(x) = a$$ is given by $$f(x) =...
4
votes
2answers
83 views

Solving the functional equation $f(x)f(y)=c\,f(\sqrt{x^{2}+y^{2}})$

Find all probability density function $f:\mathbb{R}\to\mathbb{R}$ such that there exists a constant $c\in\mathbb{R}$ for which $$f(x)f(y)=c\,f(\sqrt{x^{2}+y^{2}})\text{ for all }x,y\in\mathbb{R}\,.$$ ...
1
vote
1answer
47 views

find function $f$ such that $f(x)=xf(x-1)$ and $f(1) = 1$

Find function $f$ such that $f(x)=xf(x-1)$ and $f(1) = 1$. I can prove that there is just one function as $f$ (see Proof1). I know that there exists a pi function $\Pi(z) = \int_0^\infty e^{-t} t^z\,...
2
votes
1answer
92 views

Function $f$ with $f(x_1\cdot x_2)=f(x_1)+f(x_2)$ that is not $\log$?

Is the log-function the only function that enables the transformation of a product to a sum: $$f(x_1\cdot x_2)=f(x_1)+f(x_2)\,?$$ Yes, I can approximate the log function by a Taylor Series, but are ...
2
votes
0answers
41 views

Finding the barrier height between two local minimums of free energy [closed]

Consider the below functional $F=\int_0^L dx [d_x f(x)]^2,$ with boundary conditions $\cos 2 f(0)=\cos 2 f(L),$ $\sin 2 f(0)=\sin 2 f(L)$. The set of functions $f(x)=\frac{n \pi x}{L}$ (with integer $...
4
votes
1answer
53 views

Does $ f(pq)\times f((1-p)(1-q))=f(p(1-q))\times f(q(1-p))$ imply $f(pq)=f(p)\times f(q)$ over $[0,1]$?

Let $f(x)$ be a non-negative Lebesgue measurable (or continuous, or differentiable, or strictly monotone, as needed) function defined on $[0,1]$. Condition A: $f(x)$ satisfies $$ f(pq)\times f((1-p)(1-...
2
votes
3answers
61 views

When the function equation $f(x)f(y)=axy+b$ is solvable

Assume $a,b$ are constants. The question is whether there is a continuous function $f$ defined on $\mathbb R$ or $\mathbb C$ so that $$ f(x)f(y)=axy+b $$ Of course, such a function $f$ exists if $b=0$...
3
votes
3answers
93 views

How do I finish solving $f(x)f(2y)=f(x+4y)$?

I'm trying to solve this functional equation: $$f(x)f(2y)=f(x+4y)$$ The first thing I tried was to set $x=y=0$; then I get: $$f(0)f(0)=f(0)$$ which means that either $f(0)=0$ or we can divide the ...
3
votes
1answer
70 views

Did I do something wrong in solving this functional equation or does it have no solutions?

I was trying to solve this functional equation that I found in some papers that were given to me by a friend: $$f:\mathbb{R} \rightarrow \mathbb{R}$$ $$2f(x+y)+6y^3=f(x+2y)+x^3$$ for every $x,y \in \...
2
votes
1answer
90 views

USAMO 2018 functional equation

Find all functions $f : (0, ∞) → (0, ∞)$ such that $f(x +\frac{1}{y})+ f(y +\frac{1}{z})+ f(z + \frac{1}{x})= 1$ for all $x, y, z > 0$ with $xyz = 1$. Alright so my main question is that i first ...
0
votes
0answers
41 views

How can we Differentiate Cauchy's Formula for Repeated Integrals?

Cauchy's formula for repeated integration tells us that the $n$'th repeated integral of a function $f$, evaluated from $a$ to $x$, can be expressed as a single integral: $$f^{(-n)}(x)=\frac{1}{(n-1)!}\...
2
votes
2answers
39 views

Is $f(x) = mx + c$ the only set of solutions to $f(x + 1) - f(x) = \text{constant}$ where $m$, $c$, and $x$ are integers?

Is $f(x) = mx + c$ the only set of solutions to $$f(x + 1) - f(x) = \text{constant}$$ where $m$, $c$, and $x$ are integers? I was watching this video in Youtube (https://www.youtube.com/watch?v=...
0
votes
0answers
45 views

Is this solution unique?

Let $f(p)$ be a Lebesgue measurable function on $p\in (0,1)$ satisfying $$ f^2(p(1-p))= f(p^2)\times f((1-p)^2). $$ $f(p)=\lambda p^{\alpha}$, for some $\lambda \in \mathbb{R}$ and $ \alpha \in \...
2
votes
1answer
51 views

Are the only solutions to this implicit functional equation linear?

Let $f:(0,1] \to (-\infty,0]$ be a smooth function which is strictly negative on $(0,1)$ and satisfies $f(1)=0$. Let $\epsilon \in (0,1)$ and let $x,y:(0,\epsilon) \to (0,1]$ be smooth functions ...
4
votes
1answer
56 views

Continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $f\big(f(x)\big)=rf(x)+sx$ and $r,s \in (0, 1/2).$

I wish to find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$f\big(f(x)\big)=rf(x)+sx\quad\forall x\in\mathbb{R}\,,$$ where $r,s \in (0, 1/2)$. Here's my work so far: Let $...
1
vote
2answers
46 views

Solving $f(x)$ in a functional equation

Find of general form for $f(x)$ given $f(x)+xf\left(\displaystyle\frac{3}{x}\right)=x.$ I think we need to substitute $x$ as something else, but I'm not sure. Will $x=\displaystyle\frac{3}{x}$ help ...
1
vote
2answers
32 views

Derivative of Functional Equations

We have a functional equation with two variables, say $x$ and $y$. Our goal is to find $f'(x)$. I treated $y$ as a constant and differentiated. But then I had to take $y$ in terms of $x$ to find $f'(x)...

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