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
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 $...
12
votes
3answers
247 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 (1-2x)...
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 $x\ge{8}$
4
votes
1answer
68 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 $...
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 $b=-a$ $f(...
1
vote
0answers
47 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 ...
16
votes
4answers
324 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
26 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
114 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
116 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
61 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
29 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
47 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 mean,...
8
votes
1answer
172 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
91 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. Prove ...
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
41 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 R_{>...
4
votes
1answer
59 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
64 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
149 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
42 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
120 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
55 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
91 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, then ...
0
votes
1answer
71 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
78 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
95 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 $y\...
8
votes
1answer
130 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 $xy+yz+zx=...
1
vote
1answer
143 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
88 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
24 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
72 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
21 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
66 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
49 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
53 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
120 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
43 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 complexity. ...
7
votes
1answer
140 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
69 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
105 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 $\frac{3}{4}x$, $\left(\frac{3}{4}\right)^2x$, $\left(\...