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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16
votes
4answers
309 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 , ...
0
votes
0answers
25 views

A simple PDE related to symmetric functions

This question was posed on mathoverflow but was put on hold. So far nobody has been able to give any hint either to the solution or to why it is trivial. Disclaimer: as far as I can tell this is not ...
1
vote
1answer
106 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?
3
votes
1answer
110 views

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f (x+xy+f(y) )= (f(x)+ \frac 12 )\ (f(y)+ \frac 12 \ ).$

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $$f\left (x+xy+f(y) \right )=\left (f(x)+ \frac 12 \right )\left (f(y)+ \frac 12 \right ).$$ for every $x,y \in \mathbb R$. My work ...
2
votes
3answers
46 views

how to calculate derivative of $f_n(x)=f \circ f … \circ f(x)$? Derivative on $f \circ f_{n-1}$ or $f_{n-1} \circ f$?

Denote $f_n(x)=f \circ f ... \circ f(x)$, the $n$th power of composition multiplication of $f(x)$. Assume $f(x)$ is differentiable for any order. $f(1)=1$, $f^{'}(1)=p$, $f^{''}(1)=q$ Question: Get ...
3
votes
3answers
59 views

How to show that $f$ is a straight line?

Let $f:\mathbb R\to\mathbb R$ be continuous such that $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}~\forall~x,y\in\mathbb R.$ How to show that $f$ is a straight line?
0
votes
0answers
27 views

Functional Equation of Probability Distributions

Suppose you have a real random variable $X$ that has probability distribution $f_X$ meaning $$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$ Now assume $\Phi(f_X)$ is also a ...
1
vote
1answer
46 views

Error with the proof that all solutions to the Cauchy Functional Equation are linear

If $f(x)$ is continuous, it is known that $f(x+y)=f(x)+f(y)$ implies that $f(x)$ is linear, and non-continuous solutions are discussed in these links. (1, 2,3, 4) However, what is wrong with this ...
0
votes
0answers
26 views

Class Scatter Fitness Function Calculation

I get a fitness function for class scatter, the equation: Click here to see the Fitness Function Where : T = Transpose of Matrics Mi = class mean Mo = grand mean The equation is based from ...
8
votes
1answer
162 views

Find all $f:\mathbb {R} \rightarrow \mathbb {R}$ where $f(f(x))=f'(x)f(x)+c$

Recently, while studying calculus, I have come across multiples problems which asked the following: If $f(x)$ is a polynomial, find all $f(x)$ that $f(f(x))=f'(x)f(x)+c$, where $c$ is a constant. ...
6
votes
1answer
89 views

A Increasing Multiplicative Functional Equation where $nm$ is a cube

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be a strictly increasing function such that $$f(2)=2$$ and $$f(mn)=f(m)f(n)$$ for all positive integers $m,n$ such that $mn$ is a perfect cube. ...
1
vote
0answers
34 views

$(2+z^2) f(z) + 3 z + 4 = f(z+1)$?

Consider the equation $(2+z^2) f(z) + 3 z + 4 = f(z+1)$ Such that $f(z)$ is analytic near the positive real axis and the functional equation holds for real $z>0$. Can we express Some solutions ...
1
vote
0answers
39 views

Functional equation $f(f(f(x)f(y)))=f(x)f(y^2)$ for $f: \mathbb R \rightarrow \mathbb R$.

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that $f(f(f(x)f(y)))=f(x)f(y^2)$ for all $x, y \in \mathbb R$. I made this problem myself. It is not hard to do it for $f: \mathbb ...
4
votes
1answer
55 views

Find all function $f:\mathbb{R^+}→\mathbb{R^+}$ that satisfies $f(xy)^{xy} =f(x)^x f(y)^y$

Find all function $f:\mathbb{R^+}→\mathbb{R^+}$ that satisfies the given condition: $$f(xy)^{xy} =f(x)^x f(y)^y$$ If $f:\mathbb{R}→\mathbb{R^+}$ the question would be rather simple, as putting in ...
5
votes
0answers
59 views

Increasing function $f(x)$ such that $f(\gcd(x,y))=\gcd(f(x),f(y))$

This problem was largely inspired by this problem here. There were many counterexamples given to the problem, such as multiplicative function that maps primes to a permutation thereof. However, if ...
7
votes
2answers
88 views

Find all $f(x)$ such that $f(gcd(x,y))=gcd(f(x),f(y))$

How does one find all $f:\mathbb {Z} \rightarrow \mathbb {Z}$ that satisfies the following: $$f(gcd(x,y))=gcd(f(x),f(y))$$ I had suspected that there would be some results concerning this functional ...
2
votes
1answer
25 views

A tighter family of Markov-like inequalities

I believe there should exist tighter-than-Markov Inequalities that use the same information as the markov inequality (just expectation). Consider the proof of the markov inequality in the link below: ...
7
votes
2answers
143 views

When is it possible to have $f(x+y)=f(x)+f(y)+g(xy)$?

According to the question mentioned here, it seems that there is no function $f(x)$ such that the functional equation $$f(x+y)=f(x)+f(y)-(xy-1)^2$$ can hold. Motivated by this question, I found it ...
1
vote
1answer
39 views

New SAT Math Section: Comparing Equation of Line to Graph

This is a math question on a practice test for the New SAT that will come out in March. These questions should not go above the level of precalc. I'm posting a picture of the problem as well because ...
6
votes
2answers
113 views

Functional Equation: When $f(x+y)=f(x)+f(y)-(xy-1)^2$

How does one solve the following functional equation when $f:\mathbb{R}\rightarrow\mathbb{R}$ $$f(x+y)=f(x)+f(y)-(xy-1)^2$$ When I assumed it was a polynomial equation, it can be seen through ...
1
vote
2answers
52 views

General solution of recurrence relation [closed]

I am supposed to solve for the general solution of $f(n+2)=2(f(n+2))^2 -f(n+2)f(n)-2012$. I tried the method of generating functions but I am stuck with the power $2$ on the RHS. any other methods or ...
3
votes
1answer
89 views

(Non-continuous) solutions to $f(f(x))=kx$ and $f(x^2)=xf(x)$

Given a fixed non-zero constant $k\in\mathbb{R}$, find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $$f(f(x))=kx\quad\text{and}\quad f\left(x^2\right)=xf(x).$$ If $f$ is continuous, ...
0
votes
1answer
62 views

If $f(x-2)=x$ for all real numbers x, then what is $f(x)$?

If $f(x-2)=x$ for all real numbers x, then $f(x)=?$ I think the answer stays the same, because the given says for all real x. so is $f(x)=x$ or i am wrong?
-1
votes
2answers
75 views

If $f(xy) = f(x) + f(y)$, show that $f(.)$ can only be a logarithmic function. [duplicate]

As the question states, show that the property exhibited can only be satisfied by a logarithmic function i.e no other family of functions can satisfy the above property.
0
votes
2answers
133 views

Show that $f(x^a) = a f(x) $ and $f(x y) = f(x) + f(y)$

Let $f(x) = (x-1) \large\prod_\limits{n=1}^{\infty} \dfrac{2}{x^{2^{-n}} + 1} $ For real $x > 0$ it is easy to show that $f(x^2) = 2 f(x)$. Let $a$ be a real number. Question 1 Show that ...
4
votes
3answers
92 views

how to find all functions such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$

Find all function $f:\mathbb R\to\mathbb R$ such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$. My try: If $ x=y=0$ then $f(0)=0$ and if $x\leftarrow\frac{x+1}{2}$ and ...
8
votes
1answer
127 views

Functional Equation for $f(x-y)+f(y-z)+f(z-x)=2f(x+y+z)$

The following functional equation proved quite difficult. $1.$ $f(x)$ is a polynominal with real coeffecients. $2.$ $f(1)=2,f(2)=20$. $3.$ When for real $x,y,z$ satisfies the condition ...
1
vote
1answer
139 views

if $\ f(f(x))= x^2 + 1$ , then $\ f(6)= $?

I want to know how to solve this type of questions. How can I find $\ f(x)$ from $\ f(f(x))$ Suppose, $\ f(f(x)) = x$ , then $\ f(x)=x$ or $\ f(x)=\dfrac{(x+1)}{(x-1)}$ how to find these ...
1
vote
1answer
86 views

Find all the function that satisfy $f(x+y)+1=f(x)+f(y)$

Let function $f:R\setminus 0\to R$ such (1): $$\dfrac{f(x)}{x}=f\left(\dfrac{1}{x}\right),\forall x\neq 0$$ (2): for any $x,y$ such $$f(x)+f(y)=f(x+y)+1,\forall x+y\neq 0$$ Find $f$ Let $P(x,y)$ be ...
0
votes
0answers
23 views

Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
3
votes
2answers
65 views

find all functions satisfying the condition. [duplicate]

Find all functions $f:\mathbb{R}\to{\mathbb{R}}$ such that $f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y)$ for all $x,y\in{\mathbb{R}}$ first I put $x=y=0$ so I got $f(0)=0$ or $f(0)=2$. For the case $f(0)=2$, ...
1
vote
1answer
19 views

Showing an equation with integrals of sinus [duplicate]

I have to show the following equation: $\int_{0}^{\pi} t \cdot f(sin \; t) \; dt = \frac{\pi}{2} \int_{0}^{\pi} f(sin \; t) \; dt$ with $f : [0, 1] \rightarrow \mathbb{R}$ is continuous. I ...
4
votes
6answers
65 views

Not Understanding a specific substitution rule

I was given the question, If $f(3x+5) = x^2-1$, what is $f(2)$? I am trying to understand the reasoning why $3x+5$ is set equal to $2$.
1
vote
3answers
48 views

For which a there exists a non-constant function $a+f(x+y-xy)+f(x)f(y) \leq f(x)+f(y)$

I came across the following problem: Find for which $a \in \mathbb{R}$ there exists a non-constant function $f:(0, 1] \rightarrow \mathbb{R}$ $a+f(x+y-xy)+f(x)f(y) \leq f(x)+f(y)$ for each $x, y \in ...
2
votes
1answer
47 views

Solving a functional equation using Mobius transformations

I've done part (i) pretty easily but I've no idea about (ii). I think I want to use the earlier hint about the generators but I can't seem to get anywhere.
0
votes
0answers
113 views

Functional equation $f(f(x)+3y)=12x + f(f(y)-x)$

I found this problem on a French exchange forum : Find all the $f : \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)+3y)=12x + f(f(y)-x)$ In fact I solved the problem when $f$ is supposed to be ...
0
votes
0answers
50 views

two integral equations

I'm trying to solve the two following integral equations : 1) $y(x)=2+\int_1^x\frac{1}{ty(t)}\,dt$, $x>0$ 2) $y(x)=4+\int_0^x2t\sqrt{y(t)}\,dt$ It really looks like an ODE but I'm a bit clueless ...
2
votes
2answers
42 views

Why does the monotonicity imply $2^u < 3^v$ if and only if $3^u < 6^v$?

In the question and solution below, I am wondering how to #$7$ it says "The monotonicity of $f$" implies that $2^u < 3^v$ if and only if $3^u < 6^v$, $u,v$ being positive integers." How does ...
2
votes
1answer
48 views

If $\sin y=a\sin(x+y)$ prove $\frac{\rm d y}{\rm d x}=\frac{\sin a}{1- 2x\cos a +x^2}$

If $\sin y=a\sin(x+y)$ prove $\frac{\rm d y}{\rm d x}=\frac{\sin a}{1- 2x\cos a +x^2}$ I am not finding any proper way even to express $y$ only in terms of $x$ too which could reduce bit ...
7
votes
1answer
139 views

How to solve the functional equation $ f(f(x))=ax^2+bx+c $

Find all real numbers $a,b,c\in\mathbb{R}$ for which there exists a function $f:\mathbb{R}\to\mathbb{R}$ such that: $$ f(f(x))=ax^2+bx+c $$ for all $x\in\mathbb{R}$. The only thing I could deduce is: ...
0
votes
2answers
64 views

Sufficiency of the condition $f(x) = f(x^3)$ for $f$ to be even or constant

I've been playing around with some aspects of basic functions, and I reached a function that seemed a bit peculiar. Consider $\forall x \in \mathbb{R}$ a function $f:\mathbb{R} \rightarrow \mathbb{R}$ ...
10
votes
1answer
95 views

Solving functional equation $f(4x)-f(3x)=2x$

Given that $f(4x)-f(3x)=2x$ and that $f:\mathbb{R}\rightarrow\mathbb{R}$ is an increasing function, find $f(x)$. My thoughts so far: subtituting $\frac{3}{4}x$, $\left(\frac{3}{4}\right)^2x$, ...
8
votes
1answer
211 views

If $f(2x)=2f(x), \,f'(0)=0$ Then $f(x)=0$

Recently, when I was working on a functional equation, I encountered something like an ordinary differential equation with boundary conditions! Theorem. If the following holds for all $x \in ...
5
votes
3answers
82 views

Find all solutions to the functional equation $f(x+y)-f(y)=\cfrac{x}{y(x+y)}$

Find all solutions to the functional equation $f(x+y)-f(y)=\cfrac{x}{y(x+y)}$ I've tried the substitution technique but I didn't really get something useful. For $y=1$ I have ...
6
votes
6answers
125 views

Solving functional equation $f(x+y)=f(x)+f(y)+xy$

We are given $f(0)=0$. Then when $x+y=0$: $$0=f(-y)+f(-x)+xy$$ Can I now use $x=0$ and obtain: $$0=f(-y)?$$ Is this correct? Is there a better way to solve this equation?
2
votes
3answers
69 views

Functional equation $f(x+y)=\frac{f(x)+f(y)}{1-4f(x)f(y)}$ with $f'(1)=1/2$

Try to find the solution of the functional equation $$f(x+y)=\frac{f(x)+f(y)}{1-4f(x)f(y)}$$ with $f'(1)=1/2$.
5
votes
3answers
79 views

Differentiable function such that $f(x+y),f(x)f(y),f(x-y)$ are an arithmetic progression for all $x,y$

If $f$ is a differentiable function on $\mathbb{R}$ such that $f(x+y),f(x)f(y),f(x-y)$(taken in that order) are in arithmetic progression for all $x,y\in \mathbb{R}$ and $f(0)\neq0,$then ...
1
vote
2answers
62 views

If $f(x+y)-f(x-y)=4xy$, then is $f(0)=0$ or $f(0)=1$?

As in the title. It may be very simple, but I'm having difficulty finding the proper substitution.
1
vote
1answer
45 views

What are the conditions on this Riemann-Zeta function functional equation?

I am a huge fan of the Riemann Zeta function's functional equation: $$\large{\color\green{\zeta(x)=2^x \Gamma(1-x)\zeta(1-x)\pi^{x-1}\sin\frac{\pi x}{2}}}$$ I am curious as to what conditions on $x$ ...
1
vote
2answers
38 views

If,for all $x$,$f(x)=f(2a)^x$ and $f(x+2)=27f(x)$,then find $a$

I am finding this problem confusing : If,for all $x$,$f(x)=f(2a)^x$ and $f(x+2)=27f(x)$,then find $a$. When $x=1$ I have that $f(1)=f(2a)$ using the first identity. Then when $x=2a$ I have by ...