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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2
votes
4answers
76 views

Prove that $\lim_\limits{x\to 0}{f(x)}=0$

Let $f:\mathbb{R}\mapsto \mathbb{R}$ be a function such that: $$2\cdot f(x)-\sin(f(x))=x, \forall x\in \mathbb{R}$$ Prove that $\lim_\limits{x\to 0}{f(x)}=0$. I think I need to use the sandwich ...
1
vote
2answers
71 views

What is meaning of this question and how to solve it?

I am stuck with understanding the meaning of the question, which states: Show that $\cos(n\theta)=f_n(\cos\theta)$ for polynomials $f_n(x)$ satisfying $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
5
votes
4answers
384 views

Find a polynomial from an equality

Find all polynomials for which $$(x-8)p(2x)=8(x-1)p(x)$$ What I have done so far: for $x=8$ we get $p(8)=0$ for $x=1$ we get $p(2)=0$ So there exists a polynomial $p(x) = (x-2)(x-8)q(x)$ This is ...
17
votes
4answers
3k views

Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?

I was wondering if there is a name for a function that satisfies the conditions $f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$? Thanks and regards!
4
votes
2answers
85 views

Determine all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $xf(y)+yf(x)=(x+y)f(x^2+y^2)$ for all $x,y\in\mathbb{N}$ (contest question)

The question below is from the 2002 Canada National Olympiad. I have found one family of functions but need help in finding (or proving the non-existence) of others. Suggestions on how to improve the ...
27
votes
0answers
721 views

Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square

Question: For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions. My try: It is clear that the function $$f(x)=x$$ ...
5
votes
2answers
104 views

given $2f(x) + f(1-x) = x^2$ find $f(-5)$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. A function $f$ has property that $2f(x)+ f(1-x) = x^2$ ...
6
votes
2answers
106 views

Finding a function $h$ that satisfies $h \left ( \frac{x}{x^2+h(x)} \right )=1$

Someone gave me a random maths problem to solve: Given that $h \left ( \dfrac{x}{x^2+h(x)} \right )=1$, what is $h(x)$ The restrictions given were: $h(x) \neq constant$ $\exists \frac{dh}{dx}$ ...
6
votes
1answer
124 views

Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
0
votes
1answer
19 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
1
vote
2answers
34 views

A smooth function satisfying these functional constraints

I am looking for any function on a square $$f:[-1,1]\times [-1,1] \rightarrow [0,1]$$ with the following properties: The function $f$ is as smooth as possible, e.g. differentiable almost everywhere. ...
2
votes
1answer
37 views

Substitution with functional equations

I've found this nice introduction worksheet that I started to work through with the goal to get a better understanding of functions and finding them in equations. I've gotten so far but in this one ...
2
votes
2answers
103 views

Is a function $f:\mathbb R\to\mathbb R$ such that $f(x+y)=f(x)+f(y)$ always continuous? [duplicate]

Is there a function $f:\mathbb R\to\mathbb R$ such that $f(x+y)=f(x)+f(y)$ which is not continuous? I have proved that if it's continuous in one point $a\in\mathbb R$ then it's continuous on all ...
0
votes
1answer
68 views

Find a function $\Phi$ such that $ \Phi(x)^{T}\Phi(y)=\exp(-\|x-y\|^2/(2\sigma^2))$

It's a question from HW: Suppose we have $ \Phi:\mathbb{R}^p \to \mathbb{R}^\infty $ that satisfies: $$ \Phi\left(x\right)^{T}\Phi\left(y\right)=\exp\left(-\frac{\left\Vert x-y\right\Vert ...
4
votes
1answer
60 views

If $h(x)=f(g(f(x)))$ is bijective, what do we know about $f,g$?

Question: If $h(x)=f(g(f(x)))$ as a function $\mathbb R \rightarrow \mathbb R$ is bijective, what do we know about $f,g$, which are also functions $\mathbb R \rightarrow \mathbb R$? Is my proof ...
0
votes
2answers
39 views

Solve the functional equation $ q \, \frac{f(x+1)}{f(x)}=\frac{h(x+1)}{h(x)}. $

Let $f(x),h(x)$ be two differentiate on $\mathbb{R}$ functions, $f(0)=h(0)=1$. Solve the functional equation $$ q \, \frac{f(x+1)}{f(x)}=\frac{h(x+1)}{h(x)}, $$ here $q$ is a constant. For ...
5
votes
3answers
145 views

Can we obtain $f(y+x)=y+f(x)$ from $f(x^2+f(x)^2+x)=f(x)^2+x^2+f(x)$?

Find all function $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that $$f(m^2+f(n))=f(m)^2+n.$$ Let $P(x,y)$ be the assertion: $f(x^2+f(y))=f(x)^2+y \; \forall x,y \in \mathbb{Z}^+.$ $P(x,x)$ ...
1
vote
1answer
45 views

Easily computable function that is one-to-one and onto with 2 or more inputs

I am looking for a function that will take take n inputs and have 1 output, and knowing the 1 output and function you can get back the n inputs, without error and computationally easy in both ...
-1
votes
1answer
53 views

Result related to $f (x+y+z) =f (x)f (y)f (z)$ [closed]

If $f (x+y+z) =f (x)f (y)f (z) $for all real $x,y,z$ and $f(2)=4$ and $f'(0)=3$. Then, how to find $f(0)$ and $f'(2)$?
0
votes
0answers
65 views

How to do this estimate

If $a,b$ are two vectors in $\mathbb R^n$ satisfy the following relation \begin{equation} \frac{|a|^2}{1+(1-|a|^2)^{\frac{1}{2}}}\geq \frac{|a|^2-2\langle ...
2
votes
1answer
56 views

Solving the functional equation $2f(x)-f(1/x)=3x$

If $$2f(x)-f(1/x)=3x$$ how would I find $f(x)$? I have tried various linear and other functions but I do not know how to start this
8
votes
1answer
654 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
3
votes
1answer
33 views

D'alembert functional equation

The D'Alembert functional equation is $f(x+y)+f(x-y)=2f(x)f(y)$. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfy the functional equation for all $x,y\in\mathbb{R}$. It's well known that $f$ is of the ...
9
votes
1answer
153 views

How to prove that subadditive function has this property?

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 ...
1
vote
2answers
72 views

Follow up to previous functional equation question.

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ s.t $$f(x^2+f(y))=(x-y)^2f(x+y)$$ Putting $x=y$ yields $f(x^2+f(x))=0$. (1) Putting $y=-x$ yields $f(x^2+f(-x))=(2x)^2f(0)$. (2) By (1) $(2x)^2f(0)=0$ ...
3
votes
2answers
136 views

Solving functional equation for all real numbers.

The functional equation to be solved is $ f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$. Domain:Reals,Codomain:Reals.I found about 4 possible solutions to the equation but ran into a fundamental problem with all ...
4
votes
2answers
64 views

Functional equation on integers

Is there a function $f$ such that $$f(x,y,n)=f(x+y,y-x,n+1)$$ $$f(x,y,n)\neq f(x+1,y,n)$$ $$f(x,y,n)\neq f(x,y+1,n)$$ where $x,y$ are integers, $n$ is a positive integer and the range of $f$ is a ...
4
votes
1answer
89 views

Solve the functional equation, $f(x+y+1)= (\sqrt{f(x)} + \sqrt{f(y)})^2$

Solve the functional equation, find all real valued functions $f(x)$ s.t. $$f(x+y+1)= (\sqrt{f(x)} + \sqrt{f(y)})^2$$ given $f(0)=1$. I reached to a point that $4f(x)=f(2x+1)$
6
votes
2answers
154 views

Interesting functional equation: $f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$

Solve for the function f(x): $$f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$$ I'm not able to solve this. [For instance, I tried solving for $f(\frac{x}{x+f(x)})$, but this doesn't lead me ...
4
votes
4answers
104 views

Functional Equation: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$(x+y)(f(x)-f(y))=(x-y)f(x+y)$$ My attempt: If $x=-y \not = 0$ then $0= 2x f(0)$ so $f(0)=0$. Suppose for the sake of ...
0
votes
2answers
47 views

Solving the functional equation $f(a + b) = a + f(b)$

How would you solve: $$f(a+b)=a+f(b) ?$$ It seems similar to the Cauchy equation $$f(a+b)=f(a)+f(b),$$ but I'm not sure what to do with this. I have a feeling the only solution is $f(k)=k$ but idk. ...
2
votes
1answer
75 views

Find all function $f: \mathbb{Q} \to \mathbb{Q}$ such that $f(x+f(x)+2y)=2x+2f(f(y))$ [closed]

Find all function $f: \mathbb{Q} \to \mathbb{Q}$ such that $f(x+f(x)+2y)=2x+2f(f(y))$ for all $x,y \in \mathbb{Q}$.
0
votes
1answer
28 views

Each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$ when $f$ is additive

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an additive function such that $f(1)=0$. Then for each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$. Since $f$ is additive ...
24
votes
3answers
571 views

Do there exist functions $f$ such that $f(f(x))=x^2-x+1$ for every $x$?

My question is on the existence (or not) of a function $f:\mathbb{R}\to\mathbb{R}$ which satisfy the equation: $$f(f(x))=x^2-x+1 \text{ for every }x\in\mathbb{R}$$ Supposing that such a map do exist ...
-3
votes
0answers
40 views

A functional equation of the form f(f(x)) [duplicate]

What is the solution to this equation? $f(f(x)) = x^2-x+1$
-4
votes
2answers
103 views

Nontrivial entire $f(z)$ never equal to $0$ [closed]

I'm looking for nonconstant entire functions $f(z)$ such that $f(z)\neq 0$ for any $z$. More specifically I'm looking for nontrivial cases. So $\exp(z),\exp(z^2),...$ is not what I am looking for. ...
1
vote
3answers
55 views

$f(f(y)+1)=y+f(1)$ is bijective.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(xf(y)+x)=xy+f(x), \; \forall x,y \in \mathbb{R}.$$ I read a solution in finding this function. It states that setting $x=1$ ...
1
vote
0answers
18 views

Derivative of sum of two functional derivatives with different ranges

I have a functional of the the following form, $(o<a<1)$ : $F(g(x)) = \int_0^a \! g(x) x \, \mathrm{d}x. + \int_a^1 \! (g(x)-k)x^2 \, \mathrm{d}x. $ I want to find $ \frac{\partial ...
-1
votes
1answer
15 views

Fredholm-like equation

I have the following equation: $$\lambda=\displaystyle\int_{a}^b f(x)g(x)dx$$ Where $\lambda$ is a constant and I know the expresión for f(x). Is there any way of extracting the fucntion g(x)? I ...
0
votes
0answers
36 views

Transcendental Functional Equations

Given $f^k(x) = f \circ f \circ f\circ ...(x)$ composed $k$ times, Do the functional equations $f^k(x) = g(x)$, where $g(x)$ is a basic transcendental elementary function, for example, the inverse ...
-1
votes
2answers
126 views

If $f(x+y^3)=f(x)+[f(y)]^3$ and $f'(0)\ge0$, what can $f(10)$ be? [closed]

A real valued function satisfies the condition: $f(x+y^3)=f(x)+[f(y)]^3$ for all real $x$, $y$. If $f'(0)\ge0$ how to find $f(10)$ ?
-4
votes
2answers
87 views

Find the value of this $2f\left ( \frac{1}{2} \right )$ [closed]

IF $f(f(x))=1-x$, Find $$2f\left ( \frac{1}{2} \right )=??$$ help guys, I really tried but I couldn't.
8
votes
3answers
235 views

Solve this functional equation:

Functional equations such as this one appear only once every several years on exams, so I feel it's hard to have a sure-fire way to approach the problem, unlike, say, solving a series convergence ...
2
votes
4answers
84 views

Find all functions $\mathbb{R}^{+}\rightarrow \mathbb{R}$

Find all functions $f$: $\mathbb{R}^{+}\rightarrow \mathbb{R}$ such that $$f\left ( \frac{x}{y} \right )= f(x)+f(y)-f(x)f(y)$$ for all $x,y\in\mathbb{R}^{+}$. Here, $\mathbb{R}^{+}$, denotes the ...
3
votes
2answers
123 views

What is derivative a of a continuous function such that $f(x+y) = f(x) +f(y)$? [closed]

Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function with $f(x+y) = f(x) +f(y),\forall x,y\in \mathbb{R}.$ Find $\frac {df} {dx} $, if it is exist?
0
votes
1answer
28 views

Representation of a Functional Equation.

Let, $f:\mathbb{R}^{J+1}\to\mathbb{R}$, $g:\mathbb{R}^J\to\mathbb{R}$ and consider the following equation: $$f(x,g(x))=g(x).$$ I would like to know if the following statement is true (or find ...
28
votes
7answers
5k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but ...
0
votes
0answers
15 views

Find a derivative of equation that contains Fourier series

I need to find a derivative of follow equation $$ \left(r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] \right)({\sin\phi-k\cos\phi}) - b = 0 $$ I know the derivative of $\left(r_{0} + \sum ...
1
vote
0answers
53 views

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ for which $f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$ [duplicate]

Problem Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy $$f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$$ for all $x,y\in\mathbb{R}$. This is a contest math problem, and I have very little ...
2
votes
2answers
57 views

A difficult problem on functions

I've been trying to solve the following problem but can't wrap my head around it. Let $x$, $f(x)$, $a$, $b$ be positive integers. Furthermore, if $a > b$, then $f(a) > f(b)$. Now, if $f(f(x)) = ...