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).

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation(s) and possibly other conditions; e.g., the goal can be to find all continuous solutions. Solving the equation means finding all functions satisfying the given equation(s) and any additional conditions.This is different from the more common use of the word "equation", where the solutions are numbers. It is also different from the more common use of the word "functional", referring to a mapping from a space into the reals or complexes. For basic questions about functions use more suitable tags like or .

A common technique used in solving functional equations is finding some properties of satisfying functions by substituting variables for certain values in the equation. Proving properties of satisfying functions is also helpful - finding that a function is injective, surjective, involutive, and so on, is often a key step in finding all possible solutions. Other techniques such as exploiting symmetry, considering fixed points, and even using certain properties of domains (e.g. well-ordering) sometimes help.

Some well-known functional equations are:

More information can be found at Wikipedia.

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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 $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)) = -x$ for…
Gamma Function
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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 mathematics ranging from exercises in freshman…
Martin Sleziak
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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 $$f(x)=x$$ satisfies the given conditions, since: …
6 answers

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 of equation? If this equation doesn't have any…
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Can there be an injective function whose derivative is equivalent to its inverse function?

Let's say $f:D\to R$ is an injective function on some domain where it is also differentiable. For a real function, i.e. $D\subset\mathbb R, R\subset\mathbb R$, is it possible that $f'(x)\equiv f^{-1}(x)$? Intuitively speaking, I suspect that this is…
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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 be viewed as a "square root" of $f$ in the sense…
7 answers

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 is this: Is there a function $f$ on $\mathbb{R}$ or some…
2 answers

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: Does there exist a non-trivial, monotonically…
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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 fact the same. In other words, if we write…
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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 text of the problem does not specify further.…
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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?
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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 $n=1,f(1)=m$ $$f(f(m))=mf(m)$$ Put…
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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 come up with an answer. I would like to share this…
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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\in…
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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 be linear in $\Bbb R$. But, I'm stuck here. If…
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