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 $f(x+1/x) = x^2 +1/x^2$, find $f(x)$

Given $f(x+1/x) = x^2 +1/x^2$, find $f(x)$. Please show me the way you find it. The answer in my textbook is $f(x)=\frac{1+x^2+x^4}{x\cdot \sqrt{1-x^2}}$
youdontknowme
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Functional equation: $f(f(x,y), z) = f(x, f(y,z))$

I am curious to know about the functions $f \colon \mathbb{R}^2_{\geq 0} \to \mathbb{R}_{\geq 0}$ that satisfy the following equality. For each $\{x,y,z\} \subseteq \mathbb{R}_{\geq 0}$,$$f( f(x,y), z) = f(x, f(y,z)).$$ Examples of such functions…
user420879
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Finding $f$ such that $f(x)+f(y)=f\left(\frac{x+y}{1-xy}\right)$

Determine all functions $f$, wich are everywhere differentiable and satisfy $$f(x)+f(y)=f\left(\frac{x+y}{1-xy}\right)$$ for all real $x$ and $y$ with $x.y \ne 1$. PS.: The expression sugest some relation with $\tan(x)$ but I can't go further. Any…
Arnaldo
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Solution of $z(t+a) = h(a)z(t)$

I'm reading Howard Georgi online book on The Physics of Waves and found the following argument. Given the functional equation $$ z(t+a) = h(a)z(t) $$ he makes the following derivation (I'm citing the book): If we differentiate both sides of [the…
user519
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Find the function $f(x)$ if $f(x+2f(y))=f(x)+y+f(y)$

Let $f:\mathbb{R}\to \mathbb{R}$ such $f(x)$ at $x=0$ continuous, and for any $x,y\in \mathbb{R}$ such $$f(x+2f(y))=f(x)+y+f(y)$$ Find $f(x)$. Let $x=0,y=0$ then we have $$2f(2f(0))=f(0)++f(0)$$ Let $y=2f(0)$ then we…
math110
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Does there exist the function $f^2(x)\ge f(x+y)\left(f(x)+y \right) $

Does there exist the function $f:\mathbb R^+\rightarrow \mathbb R^+$, such that $$f^2(x)\ge f(x+y)\left(f(x)+y \right) \forall x,y \in \mathbb R^+$$ My work so far: Assume that a function exists. Then $$f(x+y)\le…
Roman83
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Functional equation: $f(x,y)=f(x+y,y)=f(x,x+y)$

Is there a nonconstant continuous function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ satisfying the functional equations $f(x,y)=f(x+y,y)=f(x,x+y)$? If the answer is yes, can we characterize all solutions? Edit: I think I got it. For every…
Luke
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Functional Equation $f\big(f(x)\big)=3x$ over natural nubers for strictly increasing $f$

If $f$ is a strictly increasing function from the naturals to the naturals, and $f\big(f(x)\big)=3x$, what are all values of $f(2012)$? I have only proven that $f(3x)=3f(x)$ but that gets nowhere :(
Andy
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Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$

Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)+2xy$. We have that $f(0) = 0$ and $f(x+1) = f(x)+f(1)+2x$ and thus $f(x+1) - f(x) = f(1)+2x$. Then we see that $f(0) = f(x)+f(-x)-2x^2 \implies f(x)+f(-x)…
user19405892
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Find all functions $f$ and $g$ for which $f(x+y) = g(xy)$.

Find all functions $f$ and $g$ for which $f(x+y) = g(xy)$. Is there anything wrong with this? We see that $f(1) =g(0)$ and $f(0) = g(0)$ so $f(1) = f(0)$. Also, $f(x) = g(0)$ and therefore $f(x) = f(1)$ and so $f$ must be constant? Similarly…
user19405892
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Question regarding the Cauchy functional equation

Is it true that, if a real function $f$ satisfies $f(x+y) = f(x) + f(y)$ and vanishes at some $k \neq 0$, then $f(x) = 0$? Over the rationals(or, allowing certain conditions like continuity or monotonicity), this is clear since it is well known…
MathematicsStudent1122
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Find $f(x)$ if for every $x$: $f(x) + f(\frac {2x-3}{x-1}) = x$

I want to find $f(x)$ if for every $x$ (except one and two): $$f(x) + f\left(\frac {2x-3}{x-1}\right) = x$$ I know that the answer goes something like $g(x)= \frac {2x-3}{x-1} $ and in conclusion $g(g(g(x)))=x$ But i don't know what to do from…
A.nishaburi
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Is the function from the Cauchy functional equation, $f(x+y)=f(x)+f(y)$ injective?

It's obviously not injective in the case of $f(x)=0$. I'm wondering if it's injective in all other cases. The other linear solutions of the form $f(x)=c\cdot x$ where $c$ is some constant are injective, what about the "wild" additive functions?
DKZU
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Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f (x+xy+f(y) )= (f(x)+ \frac 12 )\ (f(y)+ \frac 12 \ ).$

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $$f\left (x+xy+f(y) \right )=\left (f(x)+ \frac 12 \right )\left (f(y)+ \frac 12 \right ).$$ for every $x,y \in \mathbb R$. My work so far: 1) $y=-1$: $$f(f(-1))=\left(f(x)+\frac 12…
Roman83
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Functional equation $f(f(f(x)f(y)))=f(x)f(y^2)$ for $f: \mathbb R \rightarrow \mathbb R$.

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that $f(f(f(x)f(y)))=f(x)f(y^2)$ for all $x, y \in \mathbb R$. I made this problem myself. It is not hard to do it for $f: \mathbb R_{>0} \rightarrow \mathbb R_{>0}$: By symmetry in $x$…
wythagoras
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