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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12 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 ...
0
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1answer
81 views

Is this plot of heart for real?

I came across this equation while surfing through the internet.Is this for real?If it is how is it done?
9
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1answer
688 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
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0answers
26 views

How can I rearrange this formula to give it in terms of $t$? [on hold]

How can I rearrange the equation $$ e^{2t} = \frac{y^{2}(y+1)}{y-1} $$ to give it in the form $y = f(t)$?
3
votes
1answer
89 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 ...
102
votes
8answers
4k views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?

I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success: Find a continuous ...
8
votes
1answer
137 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. ...
2
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3answers
45 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 ...
70
votes
0answers
2k 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 $$f(x)=x$$ ...
2
votes
2answers
92 views

About $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$

Suppose $M$ is map from vector space $X$ to vector space $Y$, $M(0) =0$, and $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$. Does this mean that $M$ is a linear map? If not, could someone please give ...
3
votes
3answers
53 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?
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0answers
33 views

Functional equation $P(x,y)= x/(1+y) P(x-1,y) + y/(x+y)P(x,y-1)$

I am looking to solve this functional equation: $$ f(x,y)+f(y,x) =0$$ I found that $G((x-y))$ and $G(x)= - G(-x)$ are some of the solutions. Just wondering if we can find other solutions? ...
0
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0answers
24 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 ...
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1answer
40 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 ...
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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 ...
16
votes
4answers
564 views

Find all bijections $\,\,f:[0,1]\rightarrow[0,1],\,$ which satisfy $\,\,f\big(2x-f(x)\big)=x$.

A friend of mine gave me the following problem: Find all functions $f:[0,1]\to[0,1]$, which are one-to-one and onto and satisfy the following functional relation: $$ f\big(2x-f(x)\big)=x, \tag{1} $$ ...
6
votes
1answer
84 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. ...
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0answers
31 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
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0answers
33 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
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1answer
41 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 ...
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votes
0answers
23 views

Calculus (Differentiation) [duplicate]

Let $f$ be a function that satisfies the condition $f(x+y)=f(x)f(y)$ for all $x$ and $y$. a) Prove that $f(x)$ is not equal to zero. b) Assuming that $f(x)$ exists when $x$ is a real number, use the ...
5
votes
0answers
58 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 ...
6
votes
2answers
106 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 ...
7
votes
2answers
84 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 ...
7
votes
2answers
134 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
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1answer
35 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 ...
2
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1answer
20 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: ...
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2answers
49 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 ...
0
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2answers
129 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 ...
8
votes
1answer
160 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 ...
3
votes
1answer
81 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, ...
4
votes
6answers
63 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 $2$.
0
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1answer
54 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
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1answer
136 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
votes
2answers
71 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
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0answers
52 views

unique function on a hilbert space

unique function on a hilbert space How do you show that with Ω=(-1,1) there exists a unique function u such that the equations in the picture is correct?
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0answers
21 views

Extension to a continuous linear functional

Extension to a continuous linear functional May this functional be extended to a continuous linear functional on these Hilbert spaces?
4
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3answers
90 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
122 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
84 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 ...
4
votes
2answers
174 views

Solutions of the functional equation $f(2x) = \frac{f(x)+x}{2}$

How can I solve the following functional equation? $$f(2x) = \frac{f(x)+x}{2},$$ for $x \in \mathbb{R}$ with $f$ being a continuous function.
0
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0answers
18 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 ...
5
votes
3answers
119 views

Finding a unique continuous function

Let $f$ be a given continuous function on $[0,1]$. How do you prove that there is a unique continuous function $g$ on $[0,1]$ satisfying $$g(x) = \frac{1}{2}g\left(\frac{x+1}{2}\right) + f(x)$$ for ...
3
votes
2answers
63 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 $f(0)=2$, ...
2
votes
1answer
83 views

For which polynomials $P$ does there exist a polynomial $Q$ such that $P(x)=Q(x)Q'(x)$?

I am wondering for which polynomials $P$ does there exist a polynomial $Q$ such that $P(x)=Q(x)Q'(x)$? I am quite sure a complete characterisation will be very hard, but I'm looking for partial ...
1
vote
1answer
18 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 ...
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
42 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.
8
votes
2answers
1k views

Functional Equation $f(x+y)=f(x)+f(y)+f(x)f(y)$

I need to find all the continuous functions from $\mathbb R\rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)+f(x)f(y)$. I know, what I assume to be, the general way to attempt these problems, but I ...
0
votes
0answers
111 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 ...