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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Given $f(xy)=xf(y)+yf(x)$ and $f(x+y)=f(x^{2021})+f(y^{2021})$. Calculate $f(\sqrt{2020})$

Let $f:\mathbb{R}→\mathbb{R}$ such that: $$f(xy)=xf(y)+yf(x), f(x+y)=f(x^{2021})+f(y^{2021}), \forall x, y\in \mathbb{R}$$ Calculate $f(\sqrt{2020})$. So far I found out that $f(x)$ is an additive function that has the form of $f(x)=ax\log_c(x)$,…
Trường Hưng Nguyễn
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finding all the functions $ f : \mathbb N \to \mathbb N $ that satisfy $ f ( n ) + f ( m ) \mathrel {\big|} n ^ 2 + k f ( n ) m + f ( m ) ^ 2 $

$ \def \divides {\mathrel {\big|}} $ Given an integer $ k $, find all the functions $ f : \mathbb N \to \mathbb N $ that satisfy $$ f ( n ) + f ( m ) \divides n ^ 2 + k f ( n ) m + f ( m ) ^ 2 $$ for all $ m , n \in \mathbb N $. My…
user884324
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$P(x)$ is Polynomial with real coefficients & $P^n(x)$ is equal to $P(P(...(P(x))...)$ that the number of $P$ is $n$

Now prove that $Q(x)$ is divisible by polynomials $P(x)-x$ $Q(x) $=$ P^{2018} (x) $- $2P^{1439} (x)$ -$ P^{1397} (x)$ At the first i tried to Invoicing the $Q(x)$ but i got nowhere , could you help guys?
Jordan Balassa
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Different functional equations than Cauchy type? $2f(xy)=f(x)+f(y)$

Since we have solution for Cauchy functional equation, $$f(xy)= f(x)+f(y)$$ which is $C\log(x)$. However, if we have $$2f(xy)=f(x)+f(y)$$ type of functional equation, I found that its solutions are any arbitrary constant $C$, but failed to…
1729
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Functional equation $f(x^2+y^2)=f(x^2-y^2)+2xy \quad \forall\; x,y\in\Bbb R$

A double differentiable function satisfies $$f(x^2+y^2)=f(x^2-y^2)+2xy \quad \forall\; x,y\in\Bbb R$$ Given that $f(0)=0$ and $f''(0)=2$, determine $f(x)$. I tried to convert the equation into some form of Cauchy's additive function by observing…
statsisfun_sid
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Functional equation which might be related to some DE

I have been trying to solve the following functional equation $$f(x+2)+af(x+1)+bf(x)=0$$ for all real values of $x$. My guess and intuition leans towards $f(x)$ being some exponential function. I started out with $f(x)=\lambda a^x $ but ended up…
statsisfun_sid
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Functional equation with just a small part i can't understand

$h(2x)=h(x)$ This implies that $h(x)$ is a constant (as otherwise the non-identically zero polynomial $h(2x)−h(x)$ will have an infinite number of roots.) this is something I read as a part of a solution but didn't understand this part. someone pls…
Shayan Ahmad
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Cauchy's functional equation without the assumption of continuity

I have been reading the book Functional Equations and How to Solve Them by Christopher G. Small. In chapter 2 the first section about Cauchy's equation $f(x+y) = f(x)+f(y)$ has a theorem that states the following Let $f : \Bbb R → \Bbb R$ satisfy…
Bebars Hossny
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Finding all functions satisfying $ f \left( x ^ 2 + 2 y f ( x ) \right) + f \left( y ^ 2 \right) = f ( x + y ) ^ 2 $

Find all functions $ f : \mathbb R \to \mathbb R $ that satisfy $$ f \left( x ^ 2 + 2 y f ( x ) \right) + f \left( y ^ 2 \right) = f ( x + y ) ^ 2 $$ for all $ x , y \in \mathbb R $. I tried a couple of standard approaches (for example, making $ f…
aksjdnkajsnd
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If $f(a;f(b;c)) = f(a;b) + c$ and $f(a;a) = 0$ , then find $f($1.1$; -5)$

Function $f(x;y)$defined on pairs of real numbers satisfies the conditions $f(a;a) = 0$, $f(a;f(b;c)) = f(a;b) + c$ for any $a, b, c$... Find$f($1.1$; -5)$ Let me think first , $f(a;a) = 0 $ for any real numbers. Now it can be said that: =>…
Crevious
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The functional equation $\delta(n) = \delta(2n) + \kappa(n)$

I am interested in finding a function $\delta(n)$ that solves a functional equation of the form—$$\delta(n) = \delta(2n) + \kappa(n),$$for all integers $n \ge 1$ that are powers of 2, where $\kappa(n)$ is a known function that depends on $n$. This…
Peter O.
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Solving a functional equation over $\mathbb{R}^+$

How do we solve the functional equation $f : \mathbb{R}^{+}\rightarrow \mathbb{R}^+$ such that $f(x)f(y)+f(xy) = xy$?
Terry
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Functional Equation $ f ^ { f ( x ) } ( y ) = f ( x ) f ( y ) $

Find all functions $ f $ taking real numbers to positive integers, such that $$ f ^ { f ( x ) } ( y ) = f ( x ) f ( y ) $$ holds true for all real numbers $ x $ and $ y $. Here $ f ^ n ( x ) $ is the $ n $-th iteration of $ f $ applied to $ x $;…
user829751
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Can D'Alembert's functional be derived from Cauchy functionas?

Is it possible to derive D'Alembert's functional equation from Cauchy's functional equations? If so, can somebody kindly point me to a reference? Edit: Can $f(x + y) + f(x − y) = 2f(x)f(y)$ be derived from either of $f(x+y)=f(x)+f(y)$,…
Antillar Maximus
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Proof that $\frac{3-x^{4}}{8x^{2}}$ is the only solution to the functional equation $f(x) + 3f\left(\frac{1}{x}\right) = x^{2}$

How can we prove that $f(x) + 3f\left(\frac{1}{x}\right) = x^{2}$ has only one solution and no other solution and that solution is $\frac{3-x^{4}}{8x^{2}}$. I know that this is a linear function equation, but can't find any sources which say that…
Abhishek Kaushik
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