Mathematics Stack Exchange
Mathematics Stack Exchange

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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a simple functional equation $f\left(\frac{x}2\right)+f\left(\frac{x+1}2\right)=2f(x)$

we have this functional equation $$\forall x \in [0,1] ,f\left(\frac{x}2\right)+f\left(\frac{x+1}2\right)=2f(x)$$ where $f$ is a continues function on $[0,1]$. I need to prove that : $\exists x_0 \in [0,1] :\forall x \in [0,1],\ f(x)\leq f(x_0)$.…
El-Mo
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$f(xf(x)) = 2f(x)$

The question says Find all functions that satisfy $$f(xf(x)) = 2f(x)$$. By comparing the degree of the two sides of the equation, one can easily see that $f$ can not be a non-zero polynomial or a power function. The function that satisfy the…
Amanuel Getachew
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Functional equations with 3 variables

What are the general solutions of the functional equations? $$ f(x,y)+f(y,z)=\frac{1}{f(x,z)} $$ $$ f(x,y)f(y,z)f(x,z)=1 $$
Umar
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Cauchy's Functional Equation $ f ( x + y ) = f ( x ) + f ( y ) $ with the Additional Assumption $ f \left( x ^ { n + 1 } \right) = x ^ n f ( x ) $

Assume that $ n $ is a given positive integer. Determine all functions $ f : \mathbb R \to \mathbb R $ such that $$ f ( x + y ) = f ( x ) + f ( y ) \tag 0 \label 0 $$ forall $ x , y \in \mathbb R $, and $$ f \left( x ^ { n + 1 } \right) = x ^ n f (…
Mohsen Shahriari
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Finding original function from composition of function

If $f(f(x)) = x^2 + 2$, then find $f(11)$? Given that if $a>b$ then $f(a)>f(b)$ I got this question from a study group of which I am part of. There the question was described as Let $x,f(x),a,b$ be positive integers and if $a>b$ then $f(a)>f(b)$ and…
Siddarth Singh
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find all continuous functions $f(x+y)+f(y+z)+f(z+x)=f(x)+f(y)+f(z)+f(x+y+z)$

QUESTION - Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)+f(y+z)+f(z+x)=f(x)+f(y)+f(z)+f(x+y+z)$ MY TRY - i proved that $f(0)=0$ then $f_{o}$ satisfies $f_{o}(x+y)+f_{o}(x-y)=2 f_{o}(x)$ and $f_e$ satisfies…
Ishan
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Functional equation $ f(x+yf(x^2))=f(x)+xf(xy) $

I have trouble solving the following functional equation: $f:\mathbb{R}\to\mathbb{R}$ such that $f(x+yf(x^2))=f(x)+xf(xy)$. I think $f(x)=x$ is the unique solution, especially, $f(x+y)=f(x)+f(y)$. But how do we show that?
Hans
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Solve $f(m+n)+f(mn-1)=f(m)f(n)$ over $\mathbb{Z}$

if $f: \mathbb{Z} \rightarrow \mathbb{Z}$ then find all the possible solutions for $$f(m+n)+f(mn-1)=f(m)f(n)$$ These are all what i have, i really get stucked when there are no hints ,regarding the order of the functional values ,or any values at…
M Desmond
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Can someone guide me through this functional equation?

I'm not necessarily sure how to approach this problem—or whether it even has a solution—but I would like to know an example of a non-constant function that satisfies this condition: $$f(x,y,z)f(x,z,y)f(y,x,z)f(y,z,x)f(z,x,y)f(z,y,x)=1$$ Also, could…
BonInSossusvlei
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find functions $f$ such that $f(x)+f(y) = f(g(x,y))$, $g$ is given and symmetric

I want to find solutions $f$ of the following functional equation given a function $g(x,y)$, which is symmetric ($g(x,y)= g(y,x)$) and strictly monotonic $\forall x,y \in $ Reals: $f(x)+f(y) = f(g(x,y))$ An observation I have been able to make is…
MRT
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Restricted Cauchy equation on a non-dense domain

It is really well known that if $f: \mathbf{R}\to \mathbf{R}$ is continuous and $$ \forall x,y \in \mathbf{R},\,\,\,\,f(x+y)=f(x)+f(y) $$ then $f$ is linear, i.e., there exists $a \in \mathbf{R}$ such that $f(x)=ax$ for all $x \in \mathbf{R}$. More…
Paolo Leonetti
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Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(xf(x)+f(y))=x^2+y$

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y, \in \Bbb{R} $ , $f(xf(x)+f(y))=x^2+y$ We can easily get a strong condition $f(f(y))=y $ by setting $x=0$ . By this equation we know $f$ is injective and surjective. I got lost…
LOIS
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prove $f$ is bounded given $f(t^2+u)=tf(t)+f(u)$

I have been trying to solve the functional equation $f:\Bbb R \to \Bbb R$ $f(t^2+u)=tf(t)+f(u)$. So far i have managed to show that $f$ is additive i.e. $f(a+b)=f(a)+f(b)$ which means that the condition can be simplified to $f(t^2)=tf(t)$. To show…
Ben Martin
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Functional equation: $f(3x) = 3f(x)$

Find $f(x)$ if: $$f: R \to R$$ $$f(3x) = 3f(x)$$ I've tried to find $f(x)$ by differentiating: \begin{align}(f(3x))' &= (3f(x))'\\ 3f '(3x) &=3f '(x)\\ f '(3x) &= f '(x)\\ f'(3x) - f '(x) &= 0\\ (f(3x) - f(x))' &= 0\tag{$\dagger$}\\ (2f(x))' &=…
Aren Bit
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¿How to solve this functional equation: $xf(y)+yf(x)=(x+y)f(x)f(y)$? Need some help!

So I've been reading about functional equations and how to solve them. I found a pretty interesting problem (for me) but I think I need some help, some hint. I've never worked with this kind of problems. Well, the problem is this: Find all…
Aeternal
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