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.

3976 questions
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Finding every $n$ such that there exists a $n$-th degree polynomial which satisfies $f(x^2+1)={f(x)}^2+1$

I'm interested in functional equation. I've been thinking about the following functional equation: $$f(x^2+1)={f(x)}^2+1\ \ \ \cdots(\star).$$ I found several functions such as $f(x)=x, x^2+1, (x^2+1)^2+1,\cdots$. Then, I got interested the…
mathlove
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Functional Differential equation $f'(g(x))=h(x)$

A few weeks ago I found a video on my YouTube feed that had the problem $f'(e^{x^2})=e^{x^2}$ on its thumbnail. I was disappointed to find that the video was intended to fix problems students might have with the chain rule, but I tried to solve that…
Kamal Saleh
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Find all functions $f(x)$ with the property that $f(xf(y)+x)=f(x)^2+xy$, for all $x,y\in\mathbb{R}$.

The question goes like this: Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all $x,y\in\mathbb{R}$, $$f(xf(y)+x)=f(x)^2+xy$$ My Attempt so far By setting $x=0$ we get $$f(0)=f(0)^2$$ Then either $f(0)=0$ or $f(0)=1$. Lets…
Gimbrint
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Find all $f:\mathbb{Q^+}\rightarrow \mathbb{Q^+}$ so that $\forall x,y\in\mathbb{Q^+}$ : $f(f(x)^2y)=x^3f(xy)$

Find all $f:\mathbb{Q^+}\rightarrow \mathbb{Q^+}$ so that $\forall x,y\in\mathbb{Q^+}$ : $f(f(x)^2y)=x^3f(xy)$ so $f(0)=0$ when $x=y=0$ $f(f(1)^2)=f(1)$ when $x=y=1$ $f(f(x)^2)=x^3f(x)$ when $y=1$ and therefore $f$ is injective so $f(1)=1$
user1213761
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Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(n)+2f(f(n))=3n+5$

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $$f(n)+2f(f(n))=3n+5$$ In the book, the following solution is provided. For this post, only Case 1 is relevant. From $f(1)+2f(f(1)) = 8$ we deduce that $f(1)$ is an even number…
Ellie_Wong
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Prove $f(x)\cdot f\left( \frac{1}{x}\right)=1$

Given a function $f:\mathbb{R}\rightarrow \mathbb{R}$ which satisfies $$f(x\cdot f(y))= \frac{f(x)}{y}$$ $\forall x,y \in \mathbb{R},y≠0$ and is not identically zero. Prove that $$f(x)\cdot f\left( \frac{1}{x}\right)=1$$ Putting $x=0$ gives $f(0)=0$…
Chesx
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Functional equation in $R^n$

Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ be continuous with $$f(x)-f(y) = C(x,y)(x-y)$$ for all $x,y \in \mathbb{R}^n$ and a function $C = C(x,y) \colon \mathbb{R}^n \times \mathbb{R}^n \to\mathbb{R}$. Then $f(x) = ax+b$ for some $a\in…
Rooibos
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Are all functions with a constant doubling time exponential functions?

Obviously every exponential function has a constant time $\tau$ such that $f(x+\tau) = 2f(x)$ for all $x$. I am interested in the reverse problem: Let $f: \mathbb{R} \to \mathbb{R}$ be a function which has a constant doubling time (WLOG this can be…
preferred_anon
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functional equation with definite integration

If $g:\mathbb{R}-\{0\} \rightarrow \mathbb{R},g(2020)=1,g(-3)=-1$ and $g(x)\cdot g(y)=2g(xy)-g\bigg(\frac{2020}{x}\bigg)\cdot g\bigg(\frac{2020}{y}\bigg)\forall x,y\in \mathbb{R}-\{0\}$. Then value of $\displaystyle…
jacky
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Determine all the functions $f:\mathbb R\mapsto\mathbb R$ satisfies the equation $f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)\,\forall a,b\in\mathbb R $

Determine all the functions $f:\mathbb R\mapsto\mathbb R$ satisfies the equation $f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)\,\forall a,b\in\mathbb R $ Let $$P(x,y): f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)$$ $$P(0,1): f(f(1))=1+f(0)$$ $$P(1,0): f(1+f(0))=…
math_learner
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Find all function $f$: $\Bbb R^+ \rightarrow \Bbb R^+$ such that $f\left(\frac{f(x)}{y}\right) = yf(y)*f(f(x))$

Find all function $f$: $\Bbb R^+ \rightarrow \Bbb R^+$ such that $$f\left(\frac{f(x)}{y}\right) = yf(y)*f(f(x))$$ for all $x, y\in\Bbb R^+$ My attempt:- Put $y = 1$, $f(f(x)) = f(1)*f(f(x))$ $\implies f(1) = 1 \hspace{1cm} \{\because f(x) > 0\}$ Put…
user1150809
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On a functional equation that the solution is $ax^k$

Let $h:\mathbb{R}\to\mathbb{R}. $I want to prove that the unique solution of $$h\left(\frac{x}{y}\right)h(y)h\left(\frac{1}{x}\right)=h\left(\frac{y}{x}\right)h(x)h\left(\frac{1}{y}\right)$$ is $h(x)=ax^k$. My partial solution is: Taking $x=y^2$ is…
Theory Nombre
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Solve the function equation $g^2(x)-g(x+1)-\dfrac{x^2+2x-6}{4}=0$

let $g(x)\in \Bbb R$ and for any $x\in \Bbb R$ such that $$g^2(x)-g(x+1)-\dfrac{x^2+2x-6}{4}=0, g(0)=0$$ find $g(x)$ my idea let $x\longrightarrow x+1$, then we…
math110
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Solve the functional equation $f(x)f(y)=f(y)f(xf(y))+\frac{1}{xy}$

Find all mappings $f$ from $\mathbb{R}_+$ to $\mathbb{R}_+$ such that for $\forall x,y \in \mathbb{R}_+$: $$ f(x)f(y)=f(y)f(xf(y))+\frac{1}{xy} $$ Since the domain is $\mathbb{R}_+$, we can't use the trick $x=-y$ or something like this. Also I've…
27rabbit
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Follow up to previous functional equation question: $f\big(x^2+f(y)\big)=(x-y)^2f(x+y)$

Find all $f:\mathbb{R} \to \mathbb{R}$ s.t. $$f\big(x^2+f(y)\big)=(x-y)^2f(x+y)$$ Putting $x=y$ yields $$f\big(x^2+f(x)\big)=0\text.\tag1\label1$$ Putting $y=-x$ yields $$f\big(x^2+f(-x)\big)=(2x)^2f(0)\text.\tag2\label2$$ By \eqref{1},…
John Marty
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