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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98 views

Complicated real to real functional equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x)+1)^2$$ for all $x,y \in \mathbb{R}.$ So far I have proved that $f$ is bijective. How should I continue?
8
votes
3answers
214 views

Solutions for the Functional Equation $f(x^2)=f(x)^2$

Suppose a continuous function $f:[0,1]\rightarrow [{0,1}]$ satisfies the functional equation $f(x^2)=f(x)^2$. Then I conjecture that we must have $f(x)=0$ or $f(x)=x^r$ for some real number $r\geq 0$. ...
8
votes
1answer
210 views

Maps of $\mathbb{R}^3$ preserving the cross product

Given a map $\phi:\Bbb R^3 \rightarrow \Bbb R^3$ such that for all $a,b \in \Bbb R^3$: $$\phi(a \times b)=\phi(a) \times \phi(b)$$ Is $\phi$ necessarily a rotation around the origin or the map ...
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. ...
8
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1answer
1k views

Solution for exponential function's functional equation by using a definition of derivative

let $f(0)=1$ and $f'(0)=1$. and $f(x+y)=f(x)f(y)$ for $x,y\in R$. How can I found $f(x)$ by using a definition of derivative?
8
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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 ...
8
votes
1answer
183 views

A functional equation problem

Let $f$ be a function which maps $\mathbb{Q}^{+}\to\mathbb{Q}^{+}$. And it satisfies $$ \left\{ \begin{array}{l} f(x)+f\left(\frac{1}{x}\right)=1\\ f(2x)=2f(f(x)) \end{array}\right. $$ Show that ...
8
votes
2answers
343 views

Solve the following functional equation $f(xf(y))+f(yf(x))=2xy$

Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $f(xf(y))+f(yf(x)=2xy$. By putting $x=y=0$ we get $f(0)=0$ and by putting $x=y=1$ we get $f(f(1))=1$. Let $y=f(1)\Rightarrow ...
7
votes
5answers
2k views

Find polynomials such that $(x-16)p(2x)=16(x-1)p(x)$

Find all polynomials $p(x)$ such that for all $x$, we have $$(x-16)p(2x)=16(x-1)p(x)$$ I tried working out with replacing $x$ by $\frac{x}{2},\frac{x}{4},\cdots$, to have $p(2x) \to p(0)$ but then ...
7
votes
7answers
383 views

Function satisfying $x = f(f(x))$ and $x \not= f(x)$

Is there a function that would satisfy the following conditions?: $\forall x \in X, x = f(f(x))$ and $x \not= f(x)$, where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in ...
7
votes
2answers
154 views

For all $x,y\in G$ we have: $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?

Let $G$ be a group and $f : G \to G$ a function such that for all $x,y\in G$: $$f(x f(y)) = f(x) y.$$ Prove that $f$ is an isomorphism. There are two problems here: we don't know that $f$ is a ...
7
votes
4answers
686 views

Solving the functional equation $f(x+1) - f(x-1) = g(x)$

Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies $$ f(x+1) - f(x-1) = g(x) $$
7
votes
3answers
195 views

Is every function $f : \mathbb N \to \mathbb N$ a composition $f = g\circ g$?

True or wrong: For every function $f: \mathbb N \rightarrow \mathbb N$ there is a function $g: \mathbb N \rightarrow \mathbb N$ with $f=g \circ g$.
7
votes
4answers
182 views

$f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous , and $f(x+1)+f(x)=x^2$

I would appreciate if somebody could help me with the following problem: Find $f(x)$ ($f(x)$ is not Polynomial function), given that: $f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is ...
7
votes
2answers
235 views

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$ and prove they are indeed all. Is there an easy way to prove this?
7
votes
4answers
782 views

If $f(x/2)=f(x)/2$, then $f(x)=f'(0)x$

Let $f:\mathbb R \to \mathbb R$ be differentiable such that $f(x/2)=f(x)/2$ for any $x\in \mathbb R$. How can I prove that $f(x)=f'(0)x$, for any $x\in \mathbb R$? It seems easy, but I don't know why, ...
7
votes
2answers
86 views

Solving a functional equation $f(x)+f\left(\frac{1}{1-x}\right)=x$

I was given the following homework: list all functions $f:\mathbb{R}\setminus\{0,1\}\rightarrow\mathbb{R}$ such that $f(x)+f\left(\frac{1}{1-x}\right)=x$. And obviously have I no idea what should I do ...
7
votes
3answers
644 views

If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$

If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
7
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2answers
121 views

How to solve the functional equation $f\left(x^2+f(y)\right)=y+f(x)^2$

How to solve the following functional equation: Find all $f:\mathbb{R}\to\mathbb{R} $ such that: $$ f\left(x^2+f(y)\right)=y+f(x)^2 $$ Holds for every $x,y\in\mathbb{R}$. A friend gave it to me, ...
7
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3answers
916 views

Find all the functions which satisfy a given functional equation

I was given by a friend of mine the following problem. Find all the functions $f:\mathbb R\to\mathbb R$ which satisfy the following functional equation $$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.$$ We ...
7
votes
1answer
468 views

Entire function with $f(z)=\sin(f(z))$ must be constant?

I'm trying to show why an entire function with the property $f(z)= \sin(f(z))$ everywhere must be constant. Is it sufficient to say that when taking the derivatives, we will get $f'(z)=f'(z) ...
7
votes
5answers
143 views

Find all functions $f$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$. So far, I've managed to prove that if $f$ is linear, then either $f(x) = x + 1$ or $f(x) = -1$ must be ...
7
votes
1answer
492 views

How to find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$?

Can someone please show me how to: Find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$? I've tried substitiuting $x=0,1$. Can't seem to figure it out. The square on the RHS is confusing ...
7
votes
3answers
262 views

How to find the function $f$ given $f(f(x)) = xf(x)$?

I was wondering if there is a continuous function such that $f(f(x)) = xf(x)$ for every positive number $x$.
7
votes
3answers
457 views

Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
7
votes
1answer
614 views

non-continuous function satisfies $f(x+y)=f(x)+f(y)$

As mentioned in this link, it shows that For any $f$ on the real line $\mathbb{R}^1$, $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable. But how to show there exists such ...
7
votes
2answers
77 views

How to solve $\frac{f^{-1}(x)f(x)}{x}=\frac{f^{-1}(x)+f(x)}{2}$?

I've come across the following functional equation: Determine all surjective functions $f:\mathbb{R_{>0}}\to\mathbb{R_{>0}}$ which satisfy for all $x\in\mathbb{R_{>0}}$: $$ ...
7
votes
4answers
169 views

$f:\Bbb Z\to\Bbb Z$ such that $f(f(n)-2n)=2f(n)+n$ for all $n\in\Bbb Z$

Does there exists a function $f:\Bbb Z\to\Bbb Z$ such that $$f(f(n)-2n)=2f(n)+n$$ for all $n\in\Bbb Z$
7
votes
4answers
904 views

Find a solution for $f\left(\frac{1}{x}\right)+f(x+1)=x$

Title says all. If $f$ is an analytic function on the real line, and $f\left(\dfrac{1}{x}\right)+f(x+1)=x$, what, if any, is a possible solution for $f(x)$? Additionally, what are any solutions for ...
7
votes
2answers
98 views

A functional equation with inequality

Find all (at least one) functions $f\colon \mathbb{R}\to \mathbb{R}$ (or show there is none), such that $$ f(x^3+x)≤x≤f(x^3)+f(x), \quad \text{for all $x\in \mathbb{R}$}. $$ This is a problem asked ...
7
votes
2answers
104 views

Solve the functional equation $ \dfrac{f(x)}{f(y)}=f\left( \dfrac{x-y}{f(y)} \right) $

Solve the functional equation $$ \frac{f(x)}{f(y)}=f\left( \frac{x-y}{f(y)} \right), $$ here $f: \mathbb{R} \to \mathbb{R}$ and $f$ is differentiable at $x=0.$ By set $x=y$ we get $f(0)=1$. ...
7
votes
2answers
145 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 ...
7
votes
4answers
115 views

Continuity of function where $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function which satisfies $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$ is continuous at $x=0$, then it is continuous at every point of $\mathbb{R}$. ...
7
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2answers
157 views

Problem about functional equation (Bulgarian selection team test)

This problem was taken from the bulgarian selection team test for the 47th IMO and appeared in a chinese magazine, I came across it in my own training. http://www.math.ust.hk/excalibur/v10_n4.pdf ...
7
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3answers
308 views

$f(x^2) = 2f(x)$ and $f(x)$ continuous

I ran into a problem recently where I obtained the following constraint on a function. $$f(x^2) = 2f(x) \,\,\,\, \forall x \geq 0$$ and the function $f(x)$ is continuous. Can we conclude that $f(x)$ ...
7
votes
1answer
358 views

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

Suppose $f:\mathbb R\to\mathbb R$ is a strictly decreasing function which satisfies the relation $$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) ) , \quad \forall x , y \in\mathbb R $$ ...
7
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2answers
667 views

Problem on Euler's Phi function

Let $S(n)$ be $S(n)=\left\{k\;\left|\;\left\{\frac{n}{k}\right\}\right.\geq \frac{1}{2}\right\}$,where $\{x\}$ is the fractional part of $x$ Prove that : \begin{align} \sum_{k\in S(n)} ...
7
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2answers
153 views

Function such that $f(x) f(\pi/2 - x) = 1$

I'm looking for functions that are smooth ($C^\infty$) between $0 < x < \pi/2$ that satisfy the equation $$f(x)\, f(\pi/2-x) = 1$$ on the inteverval $0<x<\pi/2$. I know that the constant ...
7
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1answer
665 views

Measurable Cauchy Function is Continuous

I found this question here Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable and I want to adapt the proof that t.b had suggested. I don't know the concepts of Baire ...
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2answers
1k views

Continuous and additive implies linear

The following problem is from Golan's linear algebra book. I have posted a solution in the comments. Problem: Let $f(x):\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function satisfying ...
7
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2answers
365 views

Defining sine and cosine

We know the following are true about sine and cosine (and that they can be proven geometrically): $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ ...
7
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2answers
163 views

Continuous functions that satisfies $f(x) + f(1-x) + f(\sqrt{x^2+(1-x)}) = 0$ and $f(\frac12)=0$

$f:[0,1]\rightarrow \mathbb{R}$ is a continuous function which satisfies $$ f(x) + f(1-x) + f\left(\sqrt{x^2+(1-x)}\right) = 0 \text{ and } f\left(\tfrac12\right)=0. $$ Can someone give explicit ...
7
votes
3answers
212 views

Is there a non-constant function $f:\mathbb{R}^2 \to \mathbb{Z}/2\mathbb{Z}$ that sums to 0 on corners of squares?

A problem in the 2009 Putnam asks about functions $f:\mathbb{R}^2 \to \mathbb{R}$ such that whenever $A,B,C,D$ are corners of some square we have $f(A)+f(B)+f(C)+f(D)=0$. Without spoiling the problem ...
7
votes
1answer
105 views

Prove $f(x)=0$ when $f(2x^2-1)= f(x)\cdot 2x$

Prove that $f(x)=0$ for $$x\in[-1, 1]$$ $f-continuous$ and for all $x$: $$f(2x^2-1)= f(x)\cdot 2x$$ It is simple for integer numbers. Another fact that I've noticed that $$f(2(-x)^2-1)= f(-x)\cdot ...
7
votes
2answers
201 views

Interesting functional equation: $f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$

Solve for the function f(x): $$f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$$ I'm not able to solve this. [For instance, I tried solving for $f(\frac{x}{x+f(x)})$, but this doesn't lead me ...
7
votes
3answers
274 views

Does there exist a function such that $f(a)f(b)=f(a^2b^2)?$

Given $S=\{2,3,4,5,6,7,\cdots,n,\cdots,\} = \Bbb N_{>1}$, prove whether there exists a function $f:S\to S$, such that for any positive $a,b$: $$f(a)f(b)=f(a^2b^2),a\neq b?$$ This is 2015 ...
7
votes
1answer
162 views

Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
7
votes
1answer
562 views

How do you solve $f'(x) = f(f(x))$?

A friend told me to solve the following differential equation: $$f'(x)=f(f(x))$$ I have no idea how to solve this! This doesn't seem to be an ordinary differential equation and I can't even solve ...
7
votes
1answer
2k views

Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…)

My lecturer was talking today (in the context of probability, more specifically Kolmogorov's axioms) about the additive property of functions, namely that: $$f(x+y) = f(x) + f(y)$$ I've been trying ...
7
votes
1answer
125 views

Find all functions f with the following two properties

Let $f(x): [0, +\infty)\mapsto \mathbb{R}$ be a function such that for one $k\in [0, +\infty)$: $$f^2(x)=k^2+x\cdot f(x+k) \quad \forall x\in \{\;[0, +\infty) : x\geq k\;\}\qquad (1)$$ and ...