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 ...

learn more… | top users | synonyms

2
votes
3answers
33 views

Additive functional equation

Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$ f(x+y) = f(x) + f(y)$$ and $$ f(f(x)) = x$$ for all $x, y \in \mathbb{R}$ This is one problem involving additive functional ...
1
vote
1answer
28 views

Functional equation - Understading an easy step in my solution.

I am trying to solve the equation and find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that: $f(m+f(n))=f(f(m))+f(n)$ for all $n, m \in \mathbb{N_{0}} $. A reasonable approach to begin with ...
0
votes
1answer
17 views

What is this function of 2 variables?

Can you tell me the function f(K,N) that has the following values? For my education, please also explain how you tackled the problem. ...
1
vote
2answers
58 views

Find all functions $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, f^n(x)=-x$

I got this problem: Prove that the only function $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, \forall x\in\Bbb{R}, f^n(x)=-x$ where $f^n =f\circ ...
1
vote
1answer
23 views

Functional equation- solving techniques

I'm basically a total novice with functional equations and have some questions regarding the solving technuiqes of them. Although, i'm adware of the lack of general solving methods, I have noticed ...
2
votes
1answer
37 views

How equal are two given numbers

I have two numbers x & y for N different readings and wish to find how close they are from each other and would like to rank the reading in order of they equalness. If I were to have the ...
3
votes
1answer
32 views

Find examples of continuous functions $f:[0,1]\to[0,1]$ that satisfy $\forall x\in[0,1], f(f(x))=f(x)$ other than $f(x)=x$

I got this problem: Find examples of continuous functions $f:[0,1]\to[0,1]$ that satisfy $\forall x\in[0,1], f(f(x))=f(x)$ other than $f(x)=x$. I proved that $f([0,1])=\{x\in[0,1]|f(x)=x\}$, But I ...
0
votes
0answers
14 views

Solving functional equation $b(x)=\int b(xy)f(y)dy$

I want to prove that given a real-valued smooth function $f$, the set of functions $b$ solving $b(x)=\int_0^{\infty} b(xy)f(y)dy$ is given by linear combinations of $x^{\sigma}$ where $\sigma$ is a ...
0
votes
1answer
70 views

Function f of x - find the x [closed]

How could I solve this? I need to find the $x$. It's an mathematical function, my teacher told me to do this, but couldn't solve it. Any hints? $$f\left(x+\frac{1}{x}\right) = x^2+\frac{1}{x}$$ ...
8
votes
4answers
141 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 ...
9
votes
3answers
118 views

How find this all function $f(x^n+2f(y))=(f(x))^n+y+f(y)$

Question: Given a positive integer $n\ge 2$ . Find all functions $f:R\to R$, such that $$f(x^n+2f(y))=(f(x))^n+y+f(y)$$ let $x=0,y=0,a=f(0)$ then $$f(2f(0))=(f(0))^n+0+f(0)\Longrightarrow ...
9
votes
4answers
525 views

What functions satisfy this functional equation?

$$f(x)-g(x)=f(g(x))$$ How could I find an f(x) and g(x) that satisfy this?
5
votes
1answer
101 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
5
votes
2answers
62 views

Finding all real valued functions that satisfy $f(f(y) + xf(x)) = y + (f(x))^2$

I would like some help with finding all real valued functions that satisfy this equation: $f(f(y) + xf(x)) = y + (f(x))^2$ I tried the usual substitutions like $x = y = 0$, but my experience with ...
0
votes
0answers
26 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...
0
votes
1answer
41 views

Writing roots of f(x) as f(a) for some a

I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots ...
0
votes
0answers
18 views

Does there exist a solution of that system of functional equations?

Does there exist a non-constant rational function $f(x)$ (i. e. a ratio of two polynomials in $x$ over the reals) which simultaneously satisfies $f(x)=f(1-x)$ and $f(x)=f\left(\frac 1 x \right)$ on ...
5
votes
1answer
94 views

How find this function such $f(2010f(n)+1389)=1+1389+1389^2+1389^3+\cdots+1389^{2010}+n$

Question: Find all function: $f:N\to N$, such that $$f(2010f(n)+1389)=1+1389+1389^2+1389^3+\cdots+1389^{2010}+n,\forall n\in N$$ Maybe this is 2010 Mathematical olympiad problem.But I ...
2
votes
1answer
39 views

How to solve the following delay differential equation?

What is the solution for the following equation? $$\frac\partial{\partial q}f(s,q)= \frac s2 f(s+2,q)$$ Note, it is known that the solution for $$\frac\partial{\partial q}f(s,q)= s f(s+1,q)$$ ...
0
votes
0answers
13 views

Extract independent paramets

I have a 2-variable function which depends also on a number of parameters (6 to be exact) $f(x,y; c1, c2, c3, .. c6)$. The explicit form is quite complicated so I will not give it here. It suffices to ...
0
votes
1answer
14 views

How do I find and list compositions for (f) and (g)?

Ok, I've literally just spent the last 2 hours just to figure out two compositions problems for homework, and I've about had it. Anyone here that can help? Problem 1 $$ f(x) = 2x(2) - x -3 $$ $$ ...
1
vote
1answer
52 views

Solution of functional equation

i know the solutions of the well known Cauchy-functional-equation $f(x+y)=f(x)+f(y)$ But what does it change if i have the following form $f(x+g(y))=f(x)+f(g(y))$ ? what can i say about g? ...
0
votes
2answers
62 views

Cauchy's functional equation with polynomial

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function that satisfies $$ f(u+av)=f(u) + f(av) + P_n(u,v) $$ where $a$ is a known constant and $P_n(u,v)$ is a polynomial in $u$ and $v$ of ...
0
votes
1answer
31 views

equation I don't understand [closed]

So this teacher of mine will give 10 point to the first who solves this equation I want a interpretation (Structure) B that satisfies phi: yhanks in advance, I know I can't ask things like this but ...
2
votes
2answers
56 views

Is there such a function $f:R\rightarrow R$?

Is there such a function $f:\mathbb R\rightarrow\mathbb R$, that for any real $x$ and $y$, we have the equality: $$ \frac{f(x)+f(y)}{2}=f\left({\frac{x+y}{2}}\right)+|x+y|\;\;\;? $$
2
votes
0answers
23 views

The existence of integral equations solution for a 2-dimensional unknown function

Suppose $f(\cdot,\cdot)\in C[0,1]^2$ is a kernel. $f$ is integratble $\int_0^1\int_0^1 f(x,y)dxdy<\infty$. $a,b\in C[0,1]$ are known functions, and $z(\cdot,\cdot)\in C[0,1]^2$ is a 2-dimensional ...
5
votes
0answers
88 views

Width of the Eiffel Tower as a function of height?

In the preface of Advanced Engineering Mathematics, 2nd Ed. by Zill and Cullen, it is claimed that the function relating the width of the Eiffel Tower as to the distance from its top, $x \mapsto ...
6
votes
1answer
159 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 ...
2
votes
1answer
20 views

Characterizing functions which satisfy de Moivre's theorem

Let $f$ and $g$ be two non-zero functions $ \mathbb R \to \mathbb R $ which are continuous and differentiable everywhere. Furthermore, say that for all integer $n$: $$ (f(\theta) + i g(\theta))^n = ...
4
votes
2answers
121 views
+50

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 ...
1
vote
0answers
47 views

Solve the function from the composition [duplicate]

I have equations as follows $$f(f(x))=x^2+x$$ Then solve for $f(x)$. Can anyone give some hints about this question?
26
votes
7answers
1k views

how to get $ f(x) $ if we know $ f(f(x))=x^2+x $

how to get $ f(x) $ if we know $ f(f(x))=x^2+x $ Is there elementary function of $f(x)$ satisfy the equation?
2
votes
0answers
53 views

How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
0
votes
0answers
12 views

Rotate an implicit surface

Say I have a the implicit equation: $F(x,y,z)=(x^2+y^2+z^2+R^2-r^2)^2-4R^2(x^2+y^2)$ for $R>r>0$. Which gives me a torus laying on the XY plane. How can I modify the equation that it might lie ...
0
votes
0answers
12 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...
0
votes
3answers
40 views

How to write a summation function that counts the number of nodes in a tree?

I come from a programming background and am interested in learning how to represent some things as simple equations, as an entry into thinking mathematically. How do you represent a tree structure as ...
3
votes
1answer
111 views

Solve for $f(x)$ if $f(f(x))=6x-f(x)$

If $f: [0,\infty) \rightarrow [0,\infty)$ and $f(f(x))=6x-f(x)$ $f(x)>0$ $ \forall x \in (0,\infty) $ Find f(x)
6
votes
4answers
86 views

Continuity of function where $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function which satisfies $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$ is continuous at $x=0$, then it is continuous at every point of $\mathbb{R}$. ...
8
votes
1answer
116 views

If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$

Question Let the function $f:\mathbb R\to\mathbb R$,and such $$f(xf(y)+x)=xy+f(x)$$ Find all $f(x)$ Let $x=1,y=1$,then $$f(f(1)+1)=1+f(1)$$ let $f(1)=t$,then $$f(t+1)=1+t$$ So I guess ...
1
vote
2answers
69 views

$f$ is a linear function. If $f(x+3) + f(4x+5) = 10x$,$ f(x)=?$

Question: Suppose $f$ is a linear function. If $f(x+3) + f(4x+5) = 10x$, then $f(x)=?$ My attempts: Okay, this is what I have so far... $$f(x) + f(3) + 5(4x) + f(5)$$ $$f(5x) = 10x - f(8)$$ ...
2
votes
0answers
47 views

If $f(2x-f(x))=x$ . Find all bijective functions.

It is given that $f :[0,1] \rightarrow [0,1] $ and it is bijective. If $f(2x-f(x))=x$ , find all such f. Is my solution correct? My attempt $f(x)$ is bijective. thus there exists g(x) which is the ...
11
votes
3answers
475 views

First order differential equation involving inverse function

I am wondering if there is a way to solve a differential equation of the following form: $$\displaystyle \frac{f'(x)}{x} = \frac{1}{f^{-1}(x)} + \frac{1}{k}$$ We can assume that $f(x): [0,T] \to ...
6
votes
2answers
119 views

if the function $f(f(n))+f(n)=2n+2014$,find the $f$

let the function $f:N^{+}\to N^{+}$,and such $$f(f(n))+f(n)=2n+2014$$ Find the $f(n)$ My try: let $n=1$,then we have $$f(f(1))+f(1)=2016$$ let $f(1)=a$,then $$f(a)+a=2016$$ and let ...
0
votes
0answers
23 views

Existence of a function satisfying all the given infima

Given a function $f : \mathbb{R} \to \mathbb{R}$, we can compute its infimum over $A$ for all the Borel measurable $A \subset \mathbb{R}$. I am wondering when we can deduce in the other direction, ...
2
votes
0answers
42 views

Correct math notation to show min and max of a list of numbers and sign?

I am working on a paper that uses a calculation from DIN 15018. The calculation is only described in text and not as a math equation. I would like to display as an actual equation if possible. The ...
0
votes
1answer
24 views

Re-expressing a function

Is it possible to re-express the function $$ f(t+x_1,t+x_2,x_1,x_2)=x_1+x_2+t $$ as $f(y_1,y_2,y_3,y_4)=???$
2
votes
2answers
142 views

Solutions to functional equation $f(x_1,x_2)+g(a_1t+x_1,a_2t+x_2)+h(b_1t+x_1,b_2t+x_2) = t^2$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$, $g:\mathbb{R}^2 \rightarrow \mathbb{R}$ and $h:\mathbb{R}^2 \rightarrow \mathbb{R}$ be twice continuously differentiable functions such that, \begin{align*} ...
1
vote
2answers
71 views

Identifying the exponential function $f(x)=e^x$ from its functional equation

Prove that if $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(x)=1+xg(x)$ where $\lim_{x\to 0}g(x)=1$, then: a) $\exists f'(x)$ $\forall x$ b) $f(x)=e^x$ I would really appreciate your help.
5
votes
1answer
76 views

Expressing a recursively defined function in terms of factorials or gamma function

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = ...
2
votes
1answer
60 views

Can polynomials with degree at least 2 over $\mathbb{R}$ have finite number of solutions in $End(\mathbb{R},+)$

Consider a polynomial of degree at least 2 with all coefficients in $\mathbb{R}$. We are concern with set of solution for the polynomial in $End(\mathbb{R},+)$ - the endomorphism ring of the abelian ...