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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Functions satisfying $f(x+y)+f(x-y)=2f(x)g(y)$

Given $f,g:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f$ is not the zero function, and $\forall x\in\mathbb{R},\; |f(x)|\leq 1$, and $\forall x,y\in\mathbb{R},\; f(x+y)+f(x-y)=2f(x)g(y)$, can we claim $\forall y\in\mathbb{R},\; |g(y)|\leq 1$…
P. Fazioli
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Find all functions such that $f(1+xf(y))=yf(x+y)$ where $x,y \in R^+$

Find all functions run over positive real numbers such that $f(1+xf(y))=yf(x+y)$ where $x,y\in R^+$ MY ANSWER: Putting $x=y=0$,we get, $f(1)=0$ Putting $x=0$ we get, $f(1)=yf(y)$ or,$yf(y)=0$ or,$f(y)=0$ (since $y\ne 0$., $y \in \mathbb…
Sufaid Saleel
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If $f : \mathbb{N} \to \mathbb{N}$ and $f(x)=f(2x^2)$ find all possible solutions

If $f : \mathbb{N} \to \mathbb{N}$ and $f(x)=f(2x^2)$ find all possible solutions to the functional equation. I think it's a constant since it doesn't really fit anything. But how do I show that?
Mathejunior
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Functional equation: $f(x)f(1/x) = f(x) + f(1/x)$.

If $x \neq 0$ , find $f(x)$ if it satisfies: $f(x)f(1/x) = f(x) + f(1/x)$. I know that the answer is $f(x) = 1 \pm x^n$ where $n \in \mathbb{R}$. I don't know how to show this.
Anon
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Find all functions $f:\mathbb R\to\mathbb R$ such that $f\left(x^2+f(y)\right)=y+f(x)^2$.

Find all functions of defined on the set of all real numbers with real values, such that $$ f\left(x^2+f(y)\right) = y + f(x)^2 $$ My attempt: Putting $x = 0$, $f\big(f(y)\big) = y + f^2(0)$ and putting $y = 0$, $f\left(x^2 + f(0)\right) =…
user540593
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Finding all $f:\mathbb{R}^+\to \mathbb{R}^+$ satisfying $f(x)\cdot f\big(yf(x)\big)=f\big(y+f(x)\big)$ for all $x,y \in \mathbb{R}^+$

Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ that satisfy $$f(x) f\big( y f(x) \big) = f\big( y + f(x) \big)$$ $\forall x,y \in \mathbb{R}^+$.
Haruboy15
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Functional equation of divisibility: $m^2+f(n)\mid mf(m)+n$

I am struggling with the following issue: Find all functions $f:\mathbb{N}_{+}\to\mathbb{N}_{+} $, such that for all positive integers $m$ and $n$, there is the divisibility $$m^2+f(n)\mid mf(m)+n\text.$$ $\mathbb{N}_{+}$ stands for the set of…
TheRlee
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Some examples of functions that are their own inverse?

I'm looking for the name and some examples of functions $f$ with the following property $$f\circ f=I$$ where $I$ is the identity. This means that $f=f^{-1}$; some examples are the functions $f(n)=-n$ and $g(n)=1/n$. What are other examples of…
Garmekain
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Solve system of equations with sin, cos, tg

I am trying to solve this system of equations but without any results. How can I solve this system of equations (in real numbers)? $$\sin^2 x + \cos^2 y = \tan^2 z$$ $$\sin^2 y + \cos^2 z = \tan^2 x$$ $$\sin^2 z + \cos^2 x = \tan^2 y$$ Thanks in…
dev0experiment
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Find all $f:\mathbb{R}\to\mathbb{R}$ with $f(x+f(y))=f(x)-f(y)$

I found that $fff(y)=f(y)$ for all $y$ by replacing the $x$ in the functional equation by 0 and setting $f(0)=c$ and then some working, which in combination also show that $f(x)=c-x$ for all $x$ in the range of $f$ since $fff=f$ makes the…
Max Freiburghaus
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Does their exist a real continuous function other than $f(x)=0$ such that $f(2x) = -2f(x)$?

I have a gut feeling it doesn't exist but I'm not sure how to prove/disprove it. My attempt: Suppose there exists $a \in \mathbb{R}\setminus\left\{0\right\}$ such that $f(a) \neq 0$ . Define $x_n = \frac{a}{2^n}$ $f(x_{n+1}) = \frac{-1}{2} f(x_n)$…
Rishi
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Functional equation which should have $f(x)=x$ as the unique solution

I'd like to solve the following functional equation: \begin{equation} 2f(x)+f(1-2x)=1 \end{equation} for $x \in [0,1/2]$. I also know that $f(1/2)=1/2$. This equation arises from a problem where I know I should get $f(x)=x$ as the unique solution. I…
folouer of kaklas
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Functional equation : $ f(f(x))=bxf(x)+a$

Does there exist real number $a, b$ and onto function $f : \mathbb{R} \to \mathbb{R}$ satisfying $$ f(f(x))=bxf(x)+a$$ for all real numbes $x$ ? My attempt : Let $f(x_1)=f(x_2)$, so $f(f(x_1))=f(f(x_2))$ so $bx_1f(x_1)=bx_2f(x_2)$ then…
user403160
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A multiplicative functional equation

I was trying to solve the following problem: Find all functions $f:(0,\infty)\to\Bbb R$ satisfying $$f(x)f(y)+f\left(\frac5x\right)f\left(\frac5y\right)=2f(xy)$$ for every $x,y\in(0,\infty)$ and $f(5)=1$. It's easy to show that $f(1)=1$ (take…
Ángela Flores
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Functional equation : $f(xf(y))=yf(x)$

Find all function $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ satisfying (i) $f(xf(y))=yf(x)$ for all positive real numbers, $x, y$ (ii) $ f(x) $ is bounded function for domain interval $(1,\infty)$ Please help me check my work below. Thank you. Let…
user403160
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