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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87 views

Find all continuous functions $f$ over real numbers such that $f(x)+x = 2f(2x)$

Find all continuous functions $f$ over real numbers such that $f(x)+x = 2f(2x)$. We have $f(0) = 0$ and $f(x) = 2f(2x) - x$, but I am not sure how to convert this functional equation into something ...
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1answer
38 views

Acceleration: If I know distance, time, and initial velocity, what's acceleration and final velocity?

So I know the Initial Velocity ($V_i$), Time ($t$), and Distance ($d$). I know that $$d = V_it + \frac{1}{2} at^2$$ If I rearrange this, would acceleration $a = \dfrac{2(d - V_it)}{t^2}$ ? Then ...
5
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1answer
74 views

There is no function from $\mathbb{R} \to (0, \infty)$ satisfying $f'(x)=f(f(x))$

Problem: Prove that there is no differentiable function from $\mathbb{R} \to (0, \infty)$ satisfying $f'(x)=f(f(x))$. I could not make much progress, except for observing that any derivatives (any ...
8
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2answers
99 views

Find all continuous functions over reals such that $f(x)+f(y) = f(x+y)-xy-1$ for all $x,y \in \mathbb{R}$

Find all continuous functions over reals such that $f(x)+f(y) = f(x+y)-xy-1$ for all $x,y \in \mathbb{R}$. I saw first that $f(0) = -1$ but then I am struggling to see how to get a formula for $f(x)...
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1answer
27 views

Find all functions $f$ over reals such that $f(xy) \geq f(x+y)$ for all $x,y \in \mathbb{R}$.

Find all functions $f$ over reals such that $f(xy) \geq f(x+y)$ for all $x,y \in \mathbb{R}$. We have that $f(x) \geq f(x+1)$ and $f(0) \geq f(1)$. I am wondering how to use these conditions to ...
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2answers
48 views

Method of solving the functional equation $f(2x)=f(x)$ using Lagrange's Mean Value Theorem

A problem i have goes as follows: Let $f:\mathbb R\to\mathbb R$ be a continuous function satisfying $f(2x)=f(x),\;\forall\;x\in\mathbb R$. If $f(1)=3$, then the value of $\displaystyle \int_{-1}^1 ...
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0answers
22 views

Does this functional equation have a (unique) solution?

Does the following equation have a (unique) solution for $b_1$? \begin{equation} b_1 = \dfrac{1}{2b_1} \left\lbrace -Var(p) + Cov \left(\left[ (p + b_1 + b_0 + \alpha)^2 -4b_1(p + b_0) \right]^{0.5},...
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0answers
43 views

Is the gamma function a solution to this problem?

A real differentiable function on $\mathbb{R}^+$ satisfying $f(x) = x!$ for $x\in\mathbb{N}$ and having minimal derivative $\left|\frac{\partial f}{\partial x}\right|$ everywhere, say in the sense ...
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1answer
41 views

Functional Equation $f(f(x))=af(x)+bx$

For real numbers $a$ and $b \neq 0$, $f(x)$ satisfies the following: $$f(f(x))=af(x)+bx$$ (1) $f(x)$ is continuous and $0<a, b<\frac{1}{2}$, show that the equation $f(x)=x$ has a real root, ...
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1answer
19 views

Functional equation $g(2x )= 1/2 g(x)$

I am trying to solve functional equation for $g: \ (0, \infty) \mapsto ( 0, \infty)$ $$ g( 2x ) =\frac 12 g(x)$$ Wolfram claims, and it is intuitive, that the function is $g(x) = C \frac{1}{x}$. But ...
0
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1answer
41 views

If $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$, find $f$

Assume $f: (0, \infty) \to \mathbb{R}$ is a continuous function such that for any $x,y > 0$, $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$. Find $f$. I would work with each condition ...
3
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1answer
74 views

Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$

Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$. We have that $f(0) = 0$ and $f(x+1) = f(x)+f(1)+2x$ and thus $f(x+1) - f(x) = f(1)+2x$. Then we see ...
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0answers
35 views

Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$

Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$. What I did was first plug in $x = 1$ to get $2f(1) = 1 \implies f(1) = \frac{1}{...
5
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2answers
89 views

All-Russian Olympiad question (composite of quadratics)

($1995$, All-Russian Olympiad, $9^{th}$ Graders, Final Round) Is it possible for the equation $f(g(h(x)))=0$, where $f, g$ and $h$ are quadratic functions, to have solutions $x=1,2,...,8$ ? I'm ...
3
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1answer
35 views

Find all functions $f$ and $g$ for which $f(x+y) = g(xy)$.

Find all functions $f$ and $g$ for which $f(x+y) = g(xy)$. Is there anything wrong with this? We see that $f(1) =g(0)$ and $f(0) = g(0)$ so $f(1) = f(0)$. Also, $f(x) = g(0)$ and therefore $f(x) = ...
2
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1answer
45 views

Find all functoins $f: \mathbb R \rightarrow \mathbb R$ such that $\forall x,y \in \mathbb R f(xy+f(x))=xf(y)+f(x)$

Find all functoins $f: \mathbb R \rightarrow \mathbb R$ such that $\forall x,y \in \mathbb R$ the equality:: $$f(xy+f(x))=xf(y)+f(x)$$ My work so far: 1) $f(0)=0$; Let $f(a)=f(b)\not=0$ $x=a, y=b \...
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1answer
54 views

Find all functions $f$, defined over real numbers that satisfy $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$

Find all functions $f$, defined over real numbers that satisfy $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$. We can set $x = 0$ and $y = n$ we get $f(n) = f(0)+f(n) \implies f(0) = 0$. Then what I ...
4
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1answer
51 views

Characterize all continuous functions such that $\int_0^1 \left(f\left(\sqrt[i]{x}\right)\right)^{k - i}\,dx = {i\over k}$ [closed]

Let $k \ge 1$ be an odd integer. What are all continuous functions $f: [0, 1] \to \textbf{R}$ such that$$\int_0^1 \left(f\left(\sqrt[i]{x}\right)\right)^{k - i}\,dx = {i\over k}$$for every $i \in \{1, ...
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2answers
32 views

Prove that there are no two functions $f$ and $g$ defined over real numbers that satisfy either of the following functional equations

Prove that there are no two functions $f$ and $g$ defined over real numbers that satisfy either of the following functional equations: $f(x)g(y) = x+y$ and $f(x) + g(y) = xy$. I think we should ...
1
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2answers
43 views

If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$

Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Show that if $f$ is continuous, then $f(x) = cx$ for all $x \in \mathbb{R}.$ I find it hard ...
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2answers
47 views

Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$

Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Prove that there is a constant $c$ for which $f(x) = cx$ for all $x \in \mathbb{Q}$. We know ...
0
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1answer
25 views

Can I prove that $f(x,y)$ can be written as $g(x+y)$ under certain conditions.

I have $f(x,y):R^2\rightarrow R$. I know $f(x,y)=f(y,x)$ and $f(x+d,y)=f(x,y+d)$. Can I prove that I can express $f(x,y)$ as $g(x+y)$. This is where I got: $f(x+d,y)=f(x,y+d)$, I plug in $x=0$ Gives ...
1
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1answer
46 views

Finding a function which satisfies an equation

During my analysis course, I found this interesting problem: Let $f: A \to A$ a function such that $f(f(x))+3x=4f(x)$ for every $x \in A$, where A is a finite set of positive real numbers. Find the ...
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2answers
48 views

Functional equation $(n-1)^2 < f(n) f(f(n)) < n^2 +n$.

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$(n-1)^2 < f(n) f(f(n)) < n^2 +n$$ for all $n \in \mathbb{N}$.
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2answers
37 views

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$:$f(f(f(x)+y)+y)=x+y+f(y)$

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$: $f(f(f(x)+y)+y)=x+y+f(y)$ I got the following: (1)$f$ is injective (2) $f(0)=0$ (3)$f(f(f(x)))=x$ But then ...
1
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1answer
51 views

Let $f$ be a function from $Q$ to $Q$ satisfying $f(x+f(y))=f(x).f(y)$. Prove that $f$ is constant

Let $f$ be a function from $Q$ to $Q$ satisfying $f(x+f(y))=f(x).f(y)$. Prove that $f$ is constant I was trying to divide into 3 cases: when f(x) has a root, when f(x)>0 and when f(x)<0 .. But I ...
0
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1answer
39 views

Help finding a $C^{1}$ function, with given $C^{1}$ functions, a relation, and some additional assumptions.

I'm down to the following problem (see below) that I just need some insight on (I couldn't find anything close to help online, via other posts here, etc.). My initial thoughts were to use the Inverse ...
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1answer
19 views

Searching a formula for scaling/mapping a variable based on 3 known values

I am sending an specc'ed integer (X) between -2048, and 2048 to a synthesizer to control its tuning. When X is 0 = Tuning on Synth is 440 (default) When X is 2048 = Tuning on Synth is 546.42 When X ...
3
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1answer
50 views

Can this Delay Differential Equation (DDE) be solved in closed form?

$$\frac{\mathrm{d}\psi}{\mathrm{d}x}=\frac{(x+\mu)^2-\lambda^2}{\mu}\psi(x+\mu)$$ where $\mu\geq0$, $\lambda\in\mathbb{C}$ are constants and $\psi(x)\to0$ as $|x|\to\infty$.
3
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1answer
63 views

Functional Equation - Rational

Fing all functions $g: R \to R$ such that, $g(x+y) + g(x)g(y) = g(xy) + g(x) + g(y)$ I have shown that $g(x) = 0$ for all $x$ and $g(x) = 2$ for all $x$ are solutions. I have also show that $g(x) = ...
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3answers
51 views

Functional equation in $a,x,y$

Let $f:(0,+\infty)\rightarrow \mathbb{R}$ and $a>0$ such that $f(a)=1$. Prove that, if \begin{align*} f(x)f(y)=f\left(\frac{a}{x}\right)f\left(\frac{a}{y}\right),\quad\forall x,y>0 \end{align*} ...
4
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1answer
100 views

Prove that there is a differentiable function $f: \mathbb{R} \to \mathbb{R}$ satisfying $[f(x)]^5+f(x)+x = 0$

Prove that there is a differentiable function $f: \mathbb{R} \to \mathbb{R}$ satisfying $$[f(x)]^5+f(x)+x = 0$$ for all $x \in \mathbb{R}$. Find $f'(x)$. Seeing how this is a functional equation, I ...
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0answers
29 views

Does mathematical modeling assist in learning how to derive equations?

I am hoping someone can provide a starting point to the question: Would mathematical modeling be a good place to start learning how to derive equations and functions from a set of data? For example, ...
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1answer
42 views

Question about solutions of nonlinear functional equation

One of the basic nonlinear functional equations is the following one: $$(f(x))^{2}=xf(2x),\ \ \ x>0.$$ I found out that functions $f(x)=2^{1-x}x\exp(cx)$ form the family of solutions of this ...
15
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2answers
237 views

Find all functions $f: \mathbb N \rightarrow \mathbb N$ such that $f(n!)=f(n)!$

Find all functions $f: \mathbb N \rightarrow \mathbb N$ (where $\mathbb N$ is the set of positive integers) such that $f(n!)=f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m)-f(n)$...
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2answers
31 views

How can I determine the best relationship for 3 variables, given several data points?

What is the best way to determine the relationship for three apparently related variables? The relationship does not appear to be linear, and may follow a combination of non-linear functions. I have ...
0
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1answer
39 views

Functional equations with nowhere differentiable solutions

As an example, the functional equation $f(x+y)=f(x)f(y)$, by declaring that $f$ is continuous and differentiable, we can arrive at the unique solution $f(x)=a^x$, by first showing that $f'(x)=f(0)f(x)$...
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 ...
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5answers
92 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 ...
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1answer
25 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 $\int \limits _0 ^1 f(x) g(x) \textrm d x$...
2
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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
41 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
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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
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1answer
47 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 ...
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0answers
22 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 ...
2
votes
2answers
66 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 $f$.
0
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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
42 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 $k\in\mathbb{N}$.