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 functional equation: $f(x, y + z) = f(x, y) + f(x + y, z)$

Can anything be said about the solutions of the following functional equation? $$ f(x, y + z) = f(x, y) + f(x + y, z) $$ I don't seem to be able to find much in what I think are the standard references in these cases.
kenshin
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Finding $f(x)$ from a equation

$f(x)+f(\frac{x-1}{x}) = x+1$ is given. find $f(x)$. I did tried to change it into another form buy substituting $x$ with $\frac{x}{x-1}$ The result were $f(\frac{1}{x})+f(\frac{x}{x-1})=\frac{2x-1}{x-1}$ I don't know what to do next.
Chutchatut Sutichavengkul
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Awkward Functional Equation $f(f(x))=(x+1)f(x)$

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(f(x))=(x+1)f(x)$ for all real $x$ and $f$ attains the value $(-1)$ at some point . Find all such functions $f$.
james_bond_007
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Is there a function so that $f(f(x))=1/x$?

I tried to look for a function that fulfills this rule; $$f(f(x))=\frac{1}{x}$$ I tried a lot, and I didn't find anything. Can anyone help me find such a function? Or prove that there isn't? It doesn't have to be a real function.
76david76
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functional equation of type $f(x+f(y)+xf(y)) = y+f(x)+yf(x)$

If $f:\mathbb{R}-\{-1\}\rightarrow \mathbb{R}$ and $f$ is a differentiable function that satisfies $$f(x+f(y)+xf(y)) = y+f(x)+yf(x)\forall x,y \in \mathbb{R}-\{-1\}\;,$$ Then value of $\displaystyle 2016(1+f(2015)) = $ $\bf{My\; Try::}$ Using…
juantheron
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$f(x+f(y))=f(x-f(y))+4xf(y)$

Find all functions $f:R\rightarrow R$ which satisfy $f(x+f(y))=f(x-f(y))+4xf(y)$ $\forall x,y \in R$. I strongly suspect $0$ and $x^2+C$ to be the only solutions but, as is almost the case with functional equations, finding the set the solutions is…
Benji
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Solving for a function

How can I find a general solution to following equation, $$ f\left(\frac{1}{y}\right)=y^2 f(y). $$ I know that $f(y) = \frac{1}{1 + y^2}$ is a solution but are there more? Is there a general technique that I can read up about for problems of this…
Raj
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Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$

Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$. What I did was first plug in $x = 1$ to get $2f(1) = 1 \implies f(1) = \frac{1}{2}$. Then seeing as how this looks symmetric I…
user19405892
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$f(m + f(n)) = f(f(m)) + f(n)$

I found this one in the list of IMO'96 (3) problems and decided to have a go at it, but could not complete the solution. So $m$ and $n$ are non-negative integers and $f$ takes values in the same set: $$f(m + f(n)) = f(f(m)) + f(n)$$ Let…
Valentin
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Way to solve basic functional equations

Is there any general way to solve basic functional equations? For example we have algebraic ways to solve algebraic equations (find $x$)! But for functional equations like : $$f(x) + f(x-1) = 0$$ or, $$f(x)-f(x^2)=1$$ How does one find $f(x)$? I…
NeilRoy
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$f(a(x))=f(x)$ - functional equation

I was reading "Functional Equations and How to Solve Them" by Small and the following comment pops up without much justification on p. 13: If $a(x)$ is an involution, then $f(a(x))=f(x)$ has as solutions $f(x) = T\,[x,a(x)]$, where $T$ is an…
A B
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Show that $f$ is a Cauchy function if $|f(x + y)| = |f(x)| + |f(y)|$.

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a solution of the functional equation $$|f(x + y)| = |f(x)| + |f(y)| \quad \forall x,y \in\mathbb{R}\text.$$ Show that $f$ is an additive function.
M'smary
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Functional equation $f(x+y)f(x-y)=\big(f(x)+f(y)\big)^2-4x^2f(y)$, extended solution

The question is Find all functions $f:\mathbb R \to\mathbb R$ such that $$f(x+y)f(x-y)=\big(f(x)+f(y)\big)^2-4x^2f(y)$$ Taking $x=y=0$, we get $f(0)^2=4f(0)^2 \implies f(0)=0$. Now take $x=y$ which immediately gives $$4f(x)^2=4x^2f(x)\\ \implies…
Abhishek Bakshi
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Solve functional equation $(x+y)(f(x)-f(y))=f(x^2)-f(y^2)$

Find all real functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ so that $(x+y)(f(x)-f(y))=f(x^2)-f(y^2)$. Can someone at least find the value of $f(1)$ if it is possible, it would help me.
CryoDrakon
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Find all continuous functions $f$ on $\mathbb{R}$ satisfying $f(x)=f(\sin x)$ for all $x$.

Find all continuous functions $f$ on $\mathbb{R}$ satisfying $f(x)=f(\sin x)$ for all $x$.
Mirzodaler
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