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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Problem in solving the functional equation $f(x-y)=f(x)f(y)+g(x)g(y)$

I have the following functional equation with me: $f(x-y)=f(x)f(y)+g(x)g(y)$ $f(x)=\cos x $ and $g(x)=\sin x$ is an obvious solution but I have some trouble proving it. I have obtained the following simple results by simple substitutions Firstly…
saisanjeev
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Let $f$ be a function $f : \mathbb{N} \to \mathbb{N}$ such that $f(2x+3y)=f(x)f(y)$, determine $f$

here what I did . $$f(0)=f(0)^2$$ so $f(0)=1$ or $f(0)=0$ IF $f(0)=1$ we have $f(2y)=f(y)$ $$f(1)=f(2)=\ldots=f(2^n)=a$$ the equation $f(x)-a=0$ has infinitly many solutions , so $f(x)=a$ since f(0)=1 , $f(x)=1$. i don't know how to handle the other…
user600785
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find all the functions $f:\mathbb{R}→\mathbb{R}$ such that $f(1)=1$ and $f(xy+f(x))=xf(y)+f(x)$

here is what i did i found that $f(f(0))=f(0)$ i need to prove that f is injective so that $f(0)$ will be equal to $0$ Hence $f(f(x))=xf(0)+f(x)$ so if $f(0)=0$ and $f$ is injective $f(x)=x$ please don't give me the solution , just give me a hint to…
user600785
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A solution for $f(ax+b)=f(x)+1$

Let $a,b$ be two constant real numbers with $a\neq 0$. Can anyone give a special solution of the functional equation $f(ax+b)=f(x)+1$, where $f:\mathbb{R}\rightarrow \mathbb{R}$? Note. It is a type of the Abel functional equations, and if $a=1$,…
M.H.Hooshmand
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Are there any real-valued functions, besides logarithms, for which $f(x*y) = f(x) + f(y)$?

Is there any real-valued function, $f$, which is not a logarithm, such that $∀ x,y$ in $ℝ$ , $f(x*y) = f(x) + f(y)$? So far, all I can think of is $z$ where $z(x) = 0$ $∀ x$ in $ℝ$ EDIT: Functions having a domain of $ℝ^+$ or a domain of $ℝ$/{0} are…
Toothpick Anemone
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If $f(f(x))=-f(x)$,does it necessarily follow that $f(x)=-x$?

Is there any counterexample other than $f(x)=0$? The actual problem states: Find a function $f: \mathbb{R} \to \mathbb{R}$, such that $f(f(x+y))=x-f(y)$. By substituting $x=0$, we get $f(f(y))=-f(y)$. Is it safe to conclude that $f(y)= -y$ for all…
Jason D
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What is a commutative, associative function which is not isomorphic to addition?

I hope I got the wording for the title right, but I basically mean a smooth function on real numbers or matrices such that $f(x,y)=f(y,x)$ and $f(x,f(y,z))=f(f(x,y),z)$ where the function is not of the form $f(x,y)=g(h(x)+h(y))$. Does such a…
Gere
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Proving that a function is additive in a functional equation

I have the equation $$h(x,k(y,z))=f(y,g(y,x)+t(y,z)), \tag{*}\label{equation}$$ where $h,k,f,g,t$ are continuous real valued functions. $h,f$ are strictly monotone in their second argument. all functions are "sensitive" to each of the arguments -…
mike
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Find all real functions that satisfy $f(xy+x)+f(y)=f(xy+y)+f(x)$.

Can you please help me solve this problem? Find all real functions that satisfy the functional equation $f(xy+x)+f(y)=f(xy+y)+f(x)$. I should solve it using the following theorem: Let $f:\mathbb R\to\mathbb R$ satisfy the Hosszu functional…
nix_nicis
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Ratio's and Max Size

If I have a video of size: width: 640 height: 480 and a screen of size: width: 1280 height: 720 what is the equation to find the max width and height I can stretch the video too and keep its ratio. I can calculate the ratio like: ratio =…
Blundell
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Question about functional equations

Let $F(x)$ be the real-valued function defined for all real $x$ except for $x = 0$ and $x = 1$ and satisfying the functional equation $$F(x) + F\left(\frac{x-1}x\right) = 1+x$$ Find $F(x)$ satisfying these conditions. Write $F(x)$ as a rational…
I. Kan
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Solve $f(x^2+y+f(y))=2y+f(x)^2$ over $\mathbb{R}$

I want to find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in\mathbb{R}$: $$f(x^2+y+f(y))=2y+f(x)^2.$$ Suppose we have given a solution $f$. Then $f$ is surjective (set $x=0$). Moreover, I have shown that…
Oliver Watt
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Is the solution of the (seemingly simple) functional equation $f(f(z))=f(z-c)+c$ unique?

Let a function $f(z)$ defined for all $z\in\mathbb{C}$ (without any further restrictions) satisfy the equation: $$ f(f(z))=f(z-c)+c, $$ where $c\ne0$ is a constant. It can be easily demonstrated that the only analytic function satisfying the…
user
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Find the functions satisfying $ \frac { f ( a b ) } { a b } = \frac { f ( a + b ) } { a + b } \cdot \frac { f ( a - b ) } { a - b } $

Find all the functions $ f : \mathbb R \to\mathbb R $ such that $$ \frac { f ( a b ) } { a b } = \frac { f ( a + b ) } { a + b } \cdot \frac { f ( a - b ) } { a - b } \text . $$ I think $ f ( x ) = x $, but I don't know how to get it.
Brain123
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Constructing a functional equation that has given solution set.

Motivation was that all the functional equations that appear in Olympiad have mainly one solution. for example, $xf(x^2)f(f(y))+f(yf(x))=f(xy)(f(f(x^2))+f(f(y^2)))$ or $f(x+y)^2=2f(x)f(y)+max(f(x^2+y^2),f(x^2)+f(y^2))$ You can find infinitely many…
Simo Ryu
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