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

0
votes
1answer
29 views

Functional Equations again

Let $f$ be a differentiable function satisfying $f(x+y)=(f(x))^{\cos y}\cdot (f(y))^{\cos x}$ for all $x,y \in R$. $f(0)=1, f'(0)=\ln 2$. If $$\int_{\frac{\pi}{2}}^{\frac{17\pi}{2}} \frac{f(x)}{f(x)...
0
votes
2answers
18 views

Number of distinct values

Question: How many possible values of (a, b, c, d), with a, b, c, d real, are there such that abc = d, bcd = a, cda = b and dab = c? I tried multiplying all the four equations to get: $$(abcd)^2 = 1$$...
1
vote
1answer
42 views

Cauchy functional inequality

Given a function on a closed interval $f\colon I\subset \mathbb{R}\to \mathbb{R}$ with $$f(x+y) \leq f(x) + f(y).$$ Moreover, I know that $f$ is monotonic increasing continuous on all points except ...
1
vote
1answer
58 views

Functional Equation: $f(x^2-y^2)=xf(x)-yf(y)$

Let $\mathbb{R}$ be the set of Real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2-y^2)=xf(x)-yf(y)$$ for all pairs of real numbers $x$ and $y$. This is a problem ...
4
votes
2answers
197 views

Does there exist the function $f^2(x)\ge f(x+y)\left(f(x)+y \right) $

Does there exist the function $f:\mathbb R^+\rightarrow \mathbb R^+$, such that $$f^2(x)\ge f(x+y)\left(f(x)+y \right) \forall x,y \in \mathbb R^+$$ My work so far: Assume that a function exists....
1
vote
1answer
35 views

Judicious guess for the solution of differential equation $y''-6y'+9y= t^{3/2} e^{3t}$

$(a)$ Let $L[y]=y''-2r_1y'+r_1^2y.$ Show that $$L[e^{r_1t}v(t)]=e^{r_1t}v''(t).$$ $(b)$Find the general solution of the equation $$y''-6y'+9y= t^{3/2} e^{3t}$$ I have problems only in part $(b)$.
0
votes
0answers
35 views

What can be said about $f$ in $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$ if $g$ is known?

Suppose you have the equailty $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$, $x,y\in\mathbb{R}$, and $g$ is known. What non-trivial facts about $f$ can be deduced from this? (Assume $f(x,y)\neq f(y,x)$...
8
votes
2answers
137 views

Find all functions $f(f(f(…(f(x_1,x_2),x_3),…),x_{2016}))=x_1+x_2+…+x_{2016}$

I am trying to solve the functional equation: Find all functions $f:\mathbb R^2\rightarrow \mathbb R$ such that for all $\left \{x_1,x_2,...,x_{2016} \right \}\subset \mathbb R$: $$f(f(f(...(f(...
2
votes
2answers
72 views

continuous (on 3, 4 and 5) f is constant, if $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is continuous on 3, 4 and 5, such that $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$. Show that f is constant. I don't know what to do ...
0
votes
1answer
60 views

How to solve this functional equation: $f(1-f(x))=1-x^{9}, f(1)=0$

I have managed to guess one solution of this function : $f(x)=1-x^{3}$, but I have no idea how to prove it unique, or get other solutions. If this is not solvable, how can you prove this function ...
2
votes
1answer
26 views

If $g:\mathbb{N}\rightarrow \mathbb{R}$ and $g(m+n)+g(m-n)=2g(m)+2g(n)$ what is $g(x)$

Determine all functions $g:\mathbb{N}\rightarrow \mathbb{R}$ such that $g(1)=1$ and $$g(m+n)+g(m-n)=2g(m)+2g(n), \quad \forall m\ge n \in \mathbb{N}$$ Because of the identity $k\cdot (a+b)^2 +k\cdot ...
4
votes
1answer
58 views

A function $f$ satisfies the condition $f[f(x) - e^x] = e + 1$ for all $x \in \Bbb R$.

Let $f$ be a function such that $f[f(x) - e^x] = e + 1$ for all $x \in \Bbb R$. Find $f(\ln 2)$. I've considered two cases: $f(x) = e^x + c$, where $c$ is constant. Then $f(c) = e^c + c = e + 1$, ...
2
votes
1answer
62 views

Functional equation.

I'm trying to solve the functional equation $f(x+f(y)) = f(x)-y$ where $f : \mathbb{Z} \to \mathbb{Z}$. What I got so far is: $f$ is injective and $f(0) = 0$. Thanks in advance for your time.
1
vote
1answer
33 views

Find all function that satisfy $(f(x) + xy) \cdot f(x - 3y) + (f(y) + xy) \cdot f(3x - y) = (f(x + y))^2$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all real numbers $x$ and $y$,$$(f(x) + xy) \cdot f(x - 3y) + (f(y) + xy) \cdot f(3x - y) = (f(x + y))^2.$$
6
votes
2answers
81 views

Solution of functional equation $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$

If $x,y\in \mathbb{R}$ and $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$ and $f'(0)=0\;,$ Then $f(x)$ is $\bf{My\; Try::}$ Using $$f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow 0}\frac{f(...
1
vote
0answers
14 views

For which $a, b, $ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx $ essentially determine $f(x) $

For which values of $a, b, $ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx $ essentially determine $f(x) $. This is a generalization of Solve ...
13
votes
3answers
704 views

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
0
votes
0answers
17 views

Uniqueness of a solution to a functional equation

I have two complex-valued functions, $f$ and $g$, that satisfy the following properties. $\overline{x}$ denotes the complex conjugate of $x$ below. $$g(t)\overline{g(t+h)} = f(h) \quad \forall{t,h}\...
5
votes
1answer
113 views

If $f(x-f(y))=f(-x)+(f(y)-2x)\cdot f(-y)$ what is $f(x)$

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x-f(y))=f(-x)+(f(y)-2x)\cdot f(-y), \quad \forall x,y \in \mathbb{R}$$ It's easy to see that $f(x)=x^2$ is a function ...
0
votes
0answers
35 views

Simple almot linear functional equation

I'd like to solve functional equation: $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$ I've managed to get: $f(0)=0,f(n)=0$ for all $n\in N$; $f(\frac{1}{2})=0$; $f(-x)=-f(x)$. I'll be grateful for any help.
0
votes
1answer
20 views

Functional equation on unit square

Suppose $F$ is a continuous function defined on the unit square $[0,1]\times[0,1]$ satisfying the following properties : i) $ F(a,a)=0,$ for all $a\in[0,1],$ ii) $F(a,b)=-F(b,a)$ for all $(a,b)\in[0,...
0
votes
1answer
23 views

The functional equation and differentiability

Find all functions $f: \mathbb R\rightarrow \mathbb R$, at the same time satisfying the following two conditions: a) $f (x + yf (x)) = f (x) f (y)$ b) the function $f$ can be represented in ...
1
vote
0answers
23 views

Find all functions such that $f(m)+f(n)|m^p+n^p$

For fixed prime number $p$, find all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)+f(n)\mid m^p+n^p$ for all $m,n\in \mathbb{Z}^+$ I managed to get only that for prime $q$ we have $f(q)=q^...
3
votes
5answers
91 views

Solve the equation $1 / \cos x = \cos x + \sin x$

I'm having trouble solving the equation $$ \frac{1}{\cos x} = \cos x + \sin x $$ For what I understand I have to make the equation $= 0$ So I get $$ \frac{1}{\cos x} - \cos x -\sin x = 0 $$ Any ...
0
votes
0answers
14 views

How to Compute Discrete Intrinsic Curvature

Given a function $f$, a region $S$ where this function is twice differentiable, and the property that all of its discrete difference series centered in $S$ converge within $S$ We have that $$ f''(x)...
4
votes
0answers
68 views

Functional equation: $f(x,y)=f(x+y,y)=f(x,x+y)$

Is there a nonconstant continuous function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ satisfying the functional equations $f(x,y)=f(x+y,y)=f(x,x+y)$? If the answer is yes, can we characterize all ...
0
votes
1answer
52 views

Functional Equation $\frac {f(x)}{f(y)} \le 2^{(x-y)^2} $

The function $f$ satisfies $$ \frac{f(x)}{f(y)} \le 2^{(x-y)^2}$$ for all $x,y$ in the domain of $f$. Then $f$ can be which of the following: (A) $\sqrt{x}+x^3$ (B) $\int_0^{\sin^2 x} \sin^{-1}\...
4
votes
1answer
80 views

If $f$ is a continuous function from $R \rightarrow R$ and $f(x)=f(x+f(x))$ then prove that $f$ is constant.

If $f$ is a continuous function from $R \rightarrow R$ and $f(x)=f(x+f(x))$ then prove that $f$ is constant. I could prove that $f(x)=f(x+f(x))=..=f(x+nf(x))$ after $n$ iterations.Then , how will I ...
1
vote
3answers
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 ...
1
vote
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
votes
1answer
73 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
votes
2answers
95 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)...
1
vote
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 ...
1
vote
2answers
44 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 ...
0
votes
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},...
1
vote
0answers
40 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 ...
1
vote
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, ...
0
votes
1answer
17 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
votes
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
votes
1answer
69 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 ...
1
vote
0answers
32 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
votes
2answers
84 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
votes
1answer
34 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
votes
1answer
44 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 \...
1
vote
1answer
53 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
votes
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, ...
0
votes
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
vote
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 ...
1
vote
2answers
45 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
votes
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 ...