Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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138
votes
7answers
15k views

Find a real 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 function $f: \...
102
votes
1answer
3k 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 $$f(x)=x$$ ...
90
votes
6answers
4k views

Does a non-trivial solution exist for $f'(x)=f(f(x))$?

Does $f'(x)=f(f(x))$ have any solutions other than $f(x)=0$? I have become convinced that it does (see below), but I don't know of any way to prove this. Is there a nice method for solving this kind ...
77
votes
1answer
12k views

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...
46
votes
7answers
5k views

A function in which addition and multiplication behave the same way

Exponents have a well-known property: $$x^ax^b = x^{a+b}$$ but $$x^{a} + x^{b} \neq x^{a+b}$$ Similarly, $$\log(a) + \log(b) = \log(ab) $$ But $$\log(a)\log(b) \neq \log(ab)$$ So my question ...
46
votes
2answers
1k views

Looking for a function such that…

There was this question on one of the whiteboards at my company, and I found it intriguing. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Anyway, here it is: ...
45
votes
8answers
5k views

On the functional square root of $x^2+1$

There are some math quizzes like: find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$ If such $\phi$ exists (it does in this example), $\phi$ can ...
43
votes
2answers
670 views

Which functions satisfy $f^n(x) = f(x)^n$ for some $n \ge 2$?

Let $n$ be an integer greater than $1$. The notation $f^n$ is notoriously ambiguous: it means either the $n$-th iterate of $f$ or its $n$-th power. I was wondering when the two interpretations are in ...
42
votes
8answers
16k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the ...
41
votes
7answers
3k views

How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?

How to get $f(x)$, if we know that $f(f(x))=x^2+x$? Is there an elementary function $f(x)$ that satisfies the equation?
33
votes
4answers
3k views

thoughts about $f(f(x))=e^x$

I was thinking, inspired by mathlinks, precisely from this post, if there exists a continuous real function $f:\mathbb R\to\mathbb R$ such that $$f(f(x))=e^x.$$ However I have not still been able to ...
33
votes
3answers
1k views

How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.

Prove that there is no continuous functions $f:[0,1]\to R$, such that $$ f(x)+f(x^2)=x. $$ My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all $x\...
32
votes
6answers
1k views

How to find $f$ if $f(f(x))=\frac{x+1}{x+2}$

let $f:\mathbb R\to \mathbb R$,and such $$f(f(x))=\dfrac{x+1}{x+2}$$ Find the $f(x)$ My try I found $f(x)=\dfrac{1}{x+1}$ because when $f(x)=\dfrac{1}{x+1}$,then $$f(f(x))=f\left(\dfrac{1}{...
32
votes
3answers
1k views

Is it true that this function $f(n)=n^{13}$?

Assume strictly monotone increasing function; such that $f:N^{+}\to N^{+}$, $h$ for all $n\in N^{+}$, $$f(f(f(n)))=f(f(n))\cdot f(n)\cdot n^{2015}$$ Prove or disprove:$f(n)=n^{13}$ Put $n=...
31
votes
2answers
518 views

Is there a bijective seacucumber?

A friend defined a seacucumber as a continuous function $f:\mathbb{C}\to\mathbb{C}$ such that $f(z+1)+f(z+i)+f(z-1)+f(z-i)=0$ for all $z\in\mathbb{C}$. He wanted to know if there exists a bijective ...
30
votes
6answers
6k views

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Find all polynomials $P$ such that $$P(x^2+1)=P(x)^2+1$$
30
votes
2answers
2k views

Functional equation: what function is its inverse's reciprocal? [duplicate]

The fact that so many students confuse functional inverse notation $$f^{-1}(x)$$ with multiplicative inverse notation $$[f(x)]^{-1}$$ got me to thinking... does there exist a function whose inverse is ...
30
votes
3answers
556 views

Recreational math: If $f(f(x))=e^x$, bound the integral $\int_0^1 f(x)dx$

I've been studying functions $f:\mathbb R\to\mathbb R$ that satisfy $f(f(x))=e^x$ (or, half-iterates of the exponential function). I know that there's only one such analytic function, but it's really ...
29
votes
4answers
5k views

Prove that this function is bounded

This is an exercise from Problems from the Book by Andreescu and Dospinescu. When it was posted on AoPS a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone ...
28
votes
2answers
427 views

How to find all polynomials satisfying $P(x^2+x-4)=P^2(x)+P(x)$?

Find all polynomials with real coefficients $P(x)$ such that $$P(x^2+x-4)=P^2(x)+P(x).$$ $P=0$ is a solution. For non-constant polynomial $P$ comparing both sides gives the leading coefficient as $1$....
28
votes
3answers
713 views

Do there exist functions $f$ such that $f(f(x))=x^2-x+1$ for every $x$?

My question is on the existence (or not) of a function $f:\mathbb{R}\to\mathbb{R}$ which satisfy the equation: $$f(f(x))=x^2-x+1 \text{ for every }x\in\mathbb{R}$$ Supposing that such a map do exist ...
28
votes
3answers
868 views

$f(ax)=f(x)^2-1$, what is $f$?

Suppose $f(ax)=(f(x))^2-1$ and suppose that $f$ is analytic in some neighborhood of $x=0$. Expanding in power series, we get $a=1+\sqrt{5}$ or $1-\sqrt{5}$. We take positive $a$. If $f\neq{\rm const}$ ...
27
votes
3answers
585 views

Show that $f(x) = \frac{x}{x+1} $ is the unique solution to a functional equation.

We are given a function $f:\mathbb Q^+ \to \mathbb Q^+$such that $$f(x)+f(1/x)=1$$ and $$f(2x)=2f(f(x))$$ Find, with proof, an explicit expression for $f(x)$ for all positive rational numbers $x$. ...
26
votes
3answers
2k views

Given $f(f(x))$ can we find $f(x)$?

Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
26
votes
6answers
2k views

IMO 1987 - function such that $f(f(n))=n+1987$

Show that there is no function $f: \mathbb{N} \to \mathbb{N}$ such that $f(f(n))=n+1987, \ \forall n \in \mathbb{N}$. This is problem 4 from IMO 1987 - see, for example, AoPS.
26
votes
2answers
2k views

Additivity + Measurability $\implies$ Continuity

A function $f:\Bbb R \to \Bbb R$ is additive and Lebesgue measurable. Prove that $f$ is continuous. I know that on $\Bbb Q$, $f$ comes out to be linear. So, if $f$ is to be continuous then $f$ must ...
25
votes
2answers
744 views

$f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$

A function that satisfies both $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ for all real $x$ is known to be the identity over $\mathbb Q$, but is it also the identity over $\mathbb R$? If not, can you provide ...
25
votes
1answer
850 views

Strange functional equation: $f(x)+f(\cos(x))=x$

BACKGROUND: A while ago, I became obsessed for a period of time with the following functional equation: $$f(x)+f(\cos(x))=x$$ I am only considering the unique real analytic solution to this functional ...
24
votes
4answers
812 views
+100

If $f(x)-f^{-1}(x)=e^{x}-1$, what is $f(x)$?

$f(x)$ is an increasing, differentiable function satisfying $f(x)-f^{-1}(x)=e^{x}-1$ for every real number $x$ I couldn't figure it out whether such function $f(x)$ exists or not. And if it exists, I ...
23
votes
5answers
6k views

Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?

I was wondering if there is a name for a function that satisfies the conditions $f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$? Thanks and regards!
23
votes
2answers
6k views

$f(a+b)=f(a)+f(b)$ but $f$ is not linear

Can you show me a continuous function $f \colon \mathbb{R}^n\to\mathbb{R}^m$ that satisfies $f(a+b)=f(a)+f(b)$ but is not linear? We have that $$f(0)=f(0+0)=2f(0)\implies f(0)=0\\ f(x-x)=f(0)=f(x)+f(...
23
votes
1answer
374 views

A function whose antiderivative equals its inverse.

Does there exist a continuous function $F$ satisfying the property \begin{align} F\left(\int^x_0 F(s)\ ds\right) = x \end{align} If yes, then is the solution unique? As stated, the question is ...
23
votes
1answer
4k views

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

Find all functions $f\colon \mathbb R\rightarrow \mathbb R$ such that for all reals $x$ and $y$: $$f(f(x)f(y))+f(x+y)=f(xy).$$ It was six hours ago in IMO 2017 (problem 2). I tried the standard ...
23
votes
1answer
299 views

A possible converse to the Cayley-Hamilton theorem?

Happy new year MSE! During my holiday vacation I had an interesting idea! The Cayley-Hamilton theorem states that if $f:\mathbb C^n\to\mathbb C^n$ is a linear function, then it is a root of its own ...
22
votes
2answers
4k views

Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?

If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
22
votes
2answers
1k views

Which $f$ satisfy the equation $\,\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y\,$?

Find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all $...
22
votes
6answers
2k views

$f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(f(x))=x^2$ for all $x$?

A friend came up with this problem, and we and a few others tried to solve it. It turned out to be really hard, so one of us asked his professor. I came with him, and it took me, him and the ...
22
votes
1answer
566 views

Solution of a function equation $f(x) + f(y) = f(x + y + 2f(xy))$

Find all functions $f:\mathbb{R}_{\geq0}\rightarrow \mathbb{R}_{\geq0}$ which satisfies that for $x,y\in\mathbb{R}_{\geq0}$, $$f(x)+f(y)=f(x+y+2f(xy))$$ I spent quite some time trying to solve it but ...
22
votes
3answers
560 views

Find all functions that satisfy $f(\frac{x+4}{1-x}) + f(x) = x$

I found the following task in a book and I would be interested if someone has an idea to solve it: Find all the functions $f$ that satisfy $f(\frac{x+4}{1-x}) + f(x) = x$. My ideas: Assuming that $...
22
votes
3answers
2k views

Proving that $f(n)=n$ if $f(n+1)>f(f(n))$

How can we prove that if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function so that $f(n+1)>f(f(n))$ for all $n\in\mathbb{N}$ then $f(n)=n$ for all $n\in\mathbb{N}$?
21
votes
5answers
525 views

Class of integrals: $I(a)=\int_0^\infty \frac{dx}{e^x+ax}$

I'm investigating integrals in the form $$I(a):=\int_0^\infty \frac{dx}{e^x+ax}$$ So far, I haven't been able to find any special values other than $I(0)=1$, and I've only managed to evaluate these ...
21
votes
2answers
492 views

Solving the infinite radical $\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+…}}}}$

$$\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+\cdots}}}}$$ This is a modification on the well-known Ramanujan infinite radical, $\sqrt{1+\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}}$, except it cannot be solved by the ...
20
votes
5answers
4k views

Prove that function is constant

Prove that a function $f:\mathbb{R}\to\mathbb{R}$ which satisfies $$f\left({\frac{x+y}3}\right)=\frac{f(x)+f(y)}2$$ is a constant function. This is my solution: constant function have derivative $0$ ...
20
votes
2answers
1k views

A functional equation with no solution

Let $f:\mathbb{R}\to (0,\infty)$ be a differentiable function satisfying $$f(f(x))=f^\prime(x)$$for each $x$. Show no such function exists. I got this problem in an exam. I haven't done anything ...
20
votes
4answers
1k views

Solving functional equation $f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$

I was trying to find functions $f:(0,+\infty)\to(0,+\infty)$ satisfying the following functional equation $$ f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y)) $$ The problem is that I can't find here any reasonable ...
20
votes
1answer
715 views

Functions $f$ satisfying $ f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R}$.

How to prove that the continuous functions $f$ on $\mathbb{R}$ satisfying $$f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R},$$ are given by $$f(x)=x+a,a\in\mathbb{R}.$$ Any hints are welcome. Thanks.
20
votes
2answers
535 views

Find $f(x)$ where $ f(x)+f\left(\frac{1-x}x\right)=x$

What function satisfies $ f(x)+f\left(\frac{1-x}x\right)=x$ ?
19
votes
3answers
22k views

If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t

Let $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$. I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as $g(xy) = g(x) + g(y)$ ...
19
votes
2answers
377 views

Find all functions $f: \mathbb N \rightarrow \mathbb N$ such that $f(n!)=f(n)!$

Find all functions $f: \mathbb N \rightarrow \mathbb N$ (where $\mathbb N$ is the set of positive integers) such that $f(n!)=f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m)-f(n)$...
19
votes
4answers
1k views

very elementary proof of Maxwell's theorem

Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...