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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0
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1answer
19 views

Different type of function equation

I've spent three days revising functions, although it wasn't enough at all because I think my sources weren't good enough to allow me to become a master in the field of math. By the way, there are two ...
1
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5answers
91 views

what is the relation between $f(x+1)$ and $f(x)$?

I searched so much over math sites and google but I didn't find helpful hints and required knowledge or the specific name of this topic in function. I stuck in relation and operations on $f(x)$ which ...
1
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2answers
33 views

Functions that satisfy the identity $f\left(\frac{x}{t}\right) f\left(-\frac{y}{r}\right)=f\left(\frac{x-y}{t-r}\right)$

I am looking for function(s) which satisfy the following property: $$f\left(\frac{x}{t}\right) f\left(-\frac{y}{r}\right)=f\left(\frac{x-y}{t-r}\right)$$ I am not sure if there is any function ...
2
votes
0answers
42 views

Resolution of equation such that $f(…f(x)…)=x$

I am wondering if it exits a way to find "easely" the solutions of an equation about a function $f$ such that $$f^{(n)}(x)=x$$ where $f^{(n)}$ is the n-th composition of $f$ itself. Obviously the ...
-1
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1answer
23 views

Integration involving functional equations [closed]

Let $f$ be a function satisfying $f(x+y) = f(x) f(y)$ with $f(0) = 1$ and $g$ be a function that satisfies $f(x) + g(x) = x^2$. Then the value of the integral $\int \limits _0 ^1 f(x) g(x) \textrm d ...
2
votes
1answer
29 views

Question regarding the Cauchy functional equation

Is it true that, if a real function $f$ satisfies $f(x+y) = f(x) + f(y)$ and vanishes at some $k \neq 0$, then $f(x) = 0$? Over the rationals(or, allowing certain conditions like continuity or ...
4
votes
0answers
37 views

Functional division $\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$

As the title suggests, the problem here is: Find all functions $f:\mathbb{Z}\to\mathbb{N}$ such that, for every $x,y\in\mathbb{Z}$, we have $$\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$$ I ...
1
vote
2answers
46 views

Notation without cases? $f(x)=\begin{cases}p,&\text{if $x=p^k$}\\1,&\text{otherwise}\end{cases}$

Is there any other way to write the function $f:\Bbb N\to\Bbb N$ such that $$f(x)=\begin{cases}p,&\text{if $x=p^k$}\\1,&\text{otherwise}\end{cases}$$ when $p$ is prime and $k\in\Bbb N$?
2
votes
1answer
43 views

The Functional Equation $f(mn)=f(m)f(n)$ where $f:\mathbb{N}\rightarrow \mathbb{R}$, $f(2)=2$, and $f(m) > fn)$ if $m>n$.

The following is exercise 3.3 from Terence Tao's "Solving Mathematical Problems." Emphasis added. Suppose $f$ is a function on the positive integers which takes real values with the following ...
2
votes
2answers
62 views

Functional equation with sqrt solutions [closed]

Let $f:(0,\infty)\to(0,\infty)$ so that for all $x,y\in(0,\infty)$ we have $$f\left(\frac {x} {f(y)}\right)=\frac {x} {f(x\sqrt y)}.$$ Find function $f$.
1
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0answers
26 views

Functional equation that models trigonometric identities

Find, with proof, all continuous functions $f,g : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x)^2 + g(x)^2 = 1$ and $2f(x)g(x)=f(2x)$. I am aware that the solution pair ...
1
vote
1answer
47 views

How to solve $2f(x)+xf(\frac{1}{x})-2f(|\sqrt2 \sin(\pi(x+\frac{1}{4}))|) = 4 \cos^2 \frac{\pi x}{2}+x\cos\frac{\pi}{x}$

I saw a question today. $$2f(x)+xf(\frac{1}{x})-2f(|\sqrt2 \sin(\pi(x+\frac{1}{4}))|) = 4 \cos^2 \frac{\pi x}{2}+x\cos\frac{\pi}{x}$$ It had options like this (one or more than one may be correct): ...
1
vote
0answers
20 views

Functional equation with only two solutions?

Recently I started studying functional equations. Now I'm trying to find all solutions to the following functional equation: $$(f(2x))^3=f(4x)((f(x))^2+xf(x)).$$ Unfortunately, I was able to show only ...
1
vote
0answers
21 views

System of linear Volterra integral equation

Consider the Volterra integral equation $$ f(t) = g(t) +\int_0^t K(t,s) f(s) ds $$ where $g(t)$ is continuous on $t\in[0,T]$ and $K(t,s)$ is a weakly singular kernel. It is well known that there ...
0
votes
1answer
21 views

Functional equation: $f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$

Find an algebraic function $f:\Bbb N\to\Bbb N$ such that $$f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$$ and $$f(1)=1$$ for all $x\in\Bbb N$ I allready know two things: $f(p^k)=p$ where $p$ is prime ...
0
votes
1answer
40 views

Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that $f(x)=f(x^y)$

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$f(x)=f(x^y)$$ for all $x,y\in\mathbb{N}$. I'm not intrested in the trivial solution $f(x)=k$, where $k\in\mathbb{N}$.
92
votes
1answer
3k 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 $$f(x)=x$$ ...
1
vote
2answers
50 views

Polynomial equation

Is it true that if $P \in \mathbb{Z}[X]$, then there exists $Q \in \mathbb{Z}[X]$ such that $P(x)=Q(x+1)-Q(x)$? Can we generalize to other rings than $\mathbb{Z}$? I came up with this by ...
28
votes
4answers
4k views

Prove that this function is bounded

This is an exercise from Problems from the Book by Andreescu and Dospinescu. When it was posted on AoPS a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone ...
1
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0answers
33 views

Find $f$ such that $(f(x)-f(y))\left( f\left(\frac{x+y}{2}\right) -f(\sqrt{xy})\right) =0,\forall x,y \in (0,\infty).$

Find all continuous function $f:(0,\infty)\to\mathbb{R}$ such that $$\displaystyle (f(x)-f(y))\left( f\left(\frac{x+y}{2}\right) -f(\sqrt{xy})\right) =0,\forall x,y \in (0,\infty).$$ My try: Assume ...
9
votes
1answer
65 views

if $f(x + y) = f(x)f(y)$ is continuous, then it has to be injective.

Let $f$: $\Bbb R$ $\rightarrow$ $\Bbb R$ be a non-constant function such that $f(a + b) = f(a)f(b)$ for all real numbers $a$ and $b$. Prove that if $f(x + y) = f(x)f(y)$ is continuous, then it has ...
17
votes
1answer
1k views

Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable

Let $f:\mathbb R \rightarrow \mathbb R$, and for every $x,y\in \mathbb R$ we have $f(x+y)=f(x)+f(y)$. Show that $f$ measurable $\Leftrightarrow f$ continuous.
7
votes
1answer
683 views

Measurable Cauchy Function is Continuous

I found this question here Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable and I want to adapt the proof that t.b had suggested. I don't know the concepts of Baire ...
0
votes
0answers
32 views

Show that $f$ is continuous and hence of the form $f(x)=cx$ [duplicate]

Let $f:\Bbb R\to \Bbb R$ be additive i.e. satisfies $f(x+y)=f(x)+f(y)$ for all $x$, $y$. and Lebesgue measurable. Show that $f$ is continuous and hence of the form $f(x)=cx$. In order to show ...
1
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0answers
61 views

Solutions for differential equations of the form: $f = f' \circ f''\circ \ldots \circ f^{(n)}$

Which are the n-times differentiable real functions that fit the condition: $f = f' \circ f'' \circ \ldots \circ f^{(n)}$ ? I think I have came up with a tentative solution for $n=2$, which may ...
0
votes
0answers
25 views

Functional differential equation

I want to solve a functional differential equation of this kind $$ \int d t'' dt''' a(t,t'') b(t'',t''') \frac{\delta u(t'',t';[x])}{\delta x(t''')}+\int d t'' c(t,t'') x(t'') ...
0
votes
0answers
21 views

Solving equation over expectation

I have an expression $(1 + n EX^k)p^{-k}$ which I would like to minimize over $k$. Here $n$ and $k$ are positive numbers, while $X$ and $p$ are in $[0,1]$. Since the expectation converges absolutely, ...
0
votes
0answers
20 views

Interval between two solutions of an equation

We are in $\mathbb{R}$ and $x\geq 0$. I have an equation: $(1-\alpha)x^{\frac{1}{1-\alpha}}-W(y)x=g(y)W(y)$ Where $\alpha \in (0,1)$, y is a parameter $0<y<1$, W and g are constants in $x$ and ...
0
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0answers
30 views

Solving the functional equation F(z+1)=Q(z)F(z)

I want to solve the functional equation $F(z+1)=Q(z)F(z)$. The $F(z)$ is a matrix function. $Q(z)$ is also a matrix function. But its compoents are all rational functions. i.e. $ F(z)=\begin{matrix} ...
0
votes
0answers
11 views

Pacejka formula - extracting the curve peak

I don't know if many of you knows the Pacejka Magical Formula, but it looks like this: $D\sin(C\arctan(Bx - E(Bx-\arctan(Bx))))$ As you can see, the formula reaches a peak point(in this case at ...
4
votes
1answer
28 views

Multiplicative even and odd functions?

An even function satisfies $$ f_e(x) = f_e(-x) $$ and a odd function $$ f_o(x) = -f_o(-x) $$ Every function can be split into an even and an odd part $$ f(x) = f_e(x) + f_o(x) = ...
0
votes
0answers
22 views

How to complete this?

Let $f$ a function defined on $]0,+\infty[$ and checking : $$\forall x,y>0,\ f(xy)=f(x)+f(y)$$ Assume that $f$ is bounded on $]1-\eta,1+\eta[$($\subset]0,+\infty[$). Let $\alpha =\underset{x\in ...
0
votes
0answers
33 views

Functional analysis with KKT conditions

I want to solve an optimization problem min $F(x_{ik}) $ subject to $x \in X$. $F$ here is a function or functional that I wish to determine. I want my optimal ...
5
votes
3answers
73 views

$f(x)$ is an analytic function in $\mathbb{R}$ such that $f(-x)f(x)=1$. What else can we find out about $f(x)$?

Well, I know that there are some easy things we can say immediately: $f(0)= \pm 1$, follows immediately $f(x)=\pm 1$ is the obvious solution, so let's look for other solutions. Moreover, let's ...
0
votes
0answers
24 views

Solving a functional convolution equation

I have a very weird functional equation that I am trying to solve. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a non-negative Lebesgue integrable function such that $\int f=1$. Let $\tilde{f}(x)=f(-x)$ ...
1
vote
0answers
47 views

What does non-degenerate mean?

The context is that f is a real-valued function on a plane such that for every non-degenerate square ABCD in the plane, f(A)+f(B)+f(C)+f(D)=0.... What does this word mean here? I've found that "in ...
0
votes
0answers
48 views

Find all functions $f: \Bbb Q \rightarrow \Bbb Q$

Find all functions $f:\Bbb Q \rightarrow\Bbb Q$ that satisfy the conditions: a) $f(f(2016))=0;$ b) $f(x+y)= f(x)+f(y)+f(2016)$ for all $x,y \in \Bbb Q.$ I tried to solve the problem using the ...
0
votes
1answer
21 views

Finding a possible region where this PDE has a solution

Consider the problem $$xu_t+u_x = 0, \quad u(0,x) = \sin x.$$ We're asked to prove that the problem doesn't have a solution defined in all of $\Bbb R^2$, and to give a possible open set in $\Bbb R^2$ ...
0
votes
1answer
22 views

Find the equation of the common part of two objects

How to find the equation of the intersection curve of the ball $ x^2 + y^2 + z^2 = 4a^2 $ (1)and the cylinder $x^2+y^2=2ax(a>0)$(2)? let (1)-(2), we can get $$z^2+2ax-4a^2=0 $$ but this is not ...
2
votes
1answer
59 views

To find whether the function is one-one and onto and also to proof a function bijective.

$f$ is a function from $\mathbb{N}$ to $\mathbb{N}$. $$ \begin{align} f(x)&=x+1,\text{ if $x$ is odd} \\ &=x-1,\text{ if $x$ is even} \end{align} $$ I have proved it one-one by taking ...
4
votes
3answers
194 views

Solve differential equation $f'(z) = e^{-2} (f(z/e))^2$

I'm curious if there's a simple closed solution to the following DE and, if so, what it is. $$\begin{align} f'(z) &= e^{-2} (f(z/e))^2 \\ f(0) &= 1. \end{align}$$
12
votes
3answers
239 views

The function $f (x) = f \left (\frac x2 \right ) + f \left (\frac x2 + \frac 12\right)$

The function $f: [0,1] → \mathbb R $ satisfies the equation $$f (x) = f \left (\frac x2 \right ) + f \left (\frac x2 + \frac 12\right)$$ for every $x$ in $[0,1]$. Can we assert that $f (x) = c ...
6
votes
1answer
127 views

Prove that $f(x)=8$ for all natural numbers $x\ge{8}$

A function $f$ is such that $$f(a+b)=f(ab)$$ for all natural numbers $a,b\ge{4}$ and $f(8)=8$. Prove that $f(x)=8$ for all natural numbers $x\ge{8}$
16
votes
4answers
317 views

If $f:[0,\infty)\to [0,\infty)$ and $f(x+y)=f(x)+f(y)$ then prove that $f(x)=ax$

Let $\,f:[0,\infty)\to [0,\infty)$ be a function such that $\,f(x+y)=f(x)+f(y),\,$ for all $\,x,y\ge 0$. Prove that $\,f(x)=ax,\,$ for some constant $a$. My proof : We have , $\,f(0)=0$. Then , ...
2
votes
2answers
112 views

Find all functions $f:\mathbb R\to \mathbb R$ such that $f(a^2+b^2)=f(a^2-b^2)+f(2ab)$ for every real $a$,$b$

I guessed $f(a)=a^2$ and $f(a)=0$, but have no idea how to get to the solutions in a good way. Edit: I did what was suggested: from $a=b=0$ $f(0)=0$ The function is even, because from $b=-a$ ...
4
votes
1answer
67 views

Find $f(x)$ if for every $x$: $f(x) + f(\frac {2x-3}{x-1}) = x$

I want to find $f(x)$ if for every $x$ (except one and two): $$f(x) + f\left(\frac {2x-3}{x-1}\right) = x$$ I know that the answer goes something like $g(x)= \frac {2x-3}{x-1} $ and in conclusion ...
10
votes
4answers
843 views

How find this function $f(1+xy)=f(x)f(y)+f(x+y)$

let $f:\mathbb R\longrightarrow \mathbb R$,and for any real numbers $x,y$ have $$f(1+xy)-f(x+y)=f(x)f(y)$$ and $f(-1)\neq 0$. Find the $f(x)$ My try:let $x=y=0$,then we have $$f(1)-f(0)=[f(0)]^2$$ ...
1
vote
0answers
46 views

Is the function from the Cauchy functional equation, $f(x+y)=f(x)+f(y)$ injective?

It's obviously not injective in the case of $f(x)=0$. I'm wondering if it's injective in all other cases. The other linear solutions of the form $f(x)=c\cdot x$ where $c$ is some constant are ...
1
vote
3answers
67 views

If a function such that $f(x+y)=f(x)+f(y)$ is continuous at $0$, then it is continuous on $\mathbb R$ [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$. If $f$ is continuous at zero how can I prove that is continuous in $\mathbb{R}$.
1
vote
1answer
110 views

Is this graph plot for real?

I came across this equation while surfing through the internet.Is this for real?If it is how is it done?