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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Find continuous $f$ such that $f(x+1)=f(x)+f(\frac{1}{x})$

Find all real continuous functions that verifies : $$f(x+1)=f(x)+f\left(\frac{1}{x}\right) \ \ \ \ \ \ (x\neq 0) $$ I found this result $\forall x\neq 1 \ \ f(x)=f\left(\frac{x}{x-1} \right)$ and I tried to study the behaviour of the…
D.md
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What are the continuous solutions to the functional equation $f(x) = \tfrac{1}{2}f(x^2)+\tfrac{1}{2}f(2x-x^2)$?

The solution set is a vector space and includes all functions of the form $f(x)=ax+b$. But apart from these observations I have nothing to say about the problem. If it helps, restrict the domain of $f$ to the unit interval. The problem came up when…
Ghwosque
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$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 \right ),\forall x\in \mathbb{R},m\in \mathbb{Z^+}$

Find all the function $f:\mathbb{R}\rightarrow \mathbb{R}$ sastisfied that $f$ continuous on $\mathbb{R}$ and $$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 \right ),\forall x\in \mathbb{R},m\in…
Haruboy15
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A functional equation of two variables

Solve the following functional equation : $f:\Bbb Z \rightarrow \Bbb Z$, $f(f(x)+y)=x+f(y+2017)$ I have no prior experience with solving functional equation but still tried a bit. I set $x=y=0$ to get $f(f(0))=f(2017)$. Can we apply $f^{-1}$ both…
Epsilon zero
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Functional equation $f(f(x)-x)=f(f(x))$

Consider the following functional equation: $$f(f(x)-x)=f(f(x))$$ where $f: \mathbb{R} \rightarrow \mathbb{R}$. Obviously $f(x)$ cannot be inverted. One solution is $f(x) = k$ where $k \in \mathbb{R}$ is a constant. Are there any other solution?
Fabius Wiesner
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Finding all continuous functions such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$

I've been working on the following homework problem: Find all continuous functions $f : \mathbb{R} → [0,∞)$ such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$ for all real numbers $x, y$. The first problem I have is I am not sure what is meant by…
SSumner
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Cauchy's functional equation -- additional condition

Consider the function $f:R \to R$ $$f(x+y)=f(x)+f(y)$$ which is known as Cauchy's functional equation. I know that if $f$ is monotonic, continuous at one point, bounded, then the only solutions are $f(x)=cx,~ \forall c$ But once I successfully…
abc...
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If $f\left(f(x+1) + f\left( x + f(x)\right)\right) = x+2$ then what is $f(10)$?

Let $f : \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ be a function satisfying $f(1)=1$ and $$f\Biggl(f(x+1) + f\Bigl( x + f(x)\Bigr)\Biggr) = x+2$$ Then what is $f(10)$? My teacher gave me this problem and I'm quite not sure how to solve this and…
dionxj8
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Find all functions $f:\mathbb{R} \mapsto \mathbb{R}$ which satisfy $f(x^2+y f(z)) =x f(x) + z f(y)$

QUESTION : Find all functions $f:\mathbb{R} \mapsto \mathbb{R}$ which satisfy $f(x^2+y f(z)) =x f(x) + z f(y)$ My doubt lies in the part where I've shown injectivity. Kindly check if my proof is correct. The solution is $f(x)=x$ for every individual…
Mathejunior
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Functional equation $f(x) = f(3x) + \tanh(x)$

The equation itself: $$f(x) = f(3x) + \tanh(x)$$ So firstly I'm solving homogeneous equation: $$f(x)=f(3x)$$ so is just periodic function $\Theta(\ln x)$ with period $\ln 3$. So: $$F(x) = \Theta(\ln x) + \hat{f}(x)$$where $\hat{f}(x)$ is the…
Mikhail Tikhonov
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Functional equation $f\left(\frac{x+y}{2}\right)+f\left(\frac{2xy}{x+y}\right)=f(x)+f(y)$ implies $2f(\sqrt{xy})=f(x)+f(y)$

Prove that if the function $f$ is defined on the set of positive real numbers, its values are real, and $f$ satisfies the equation $$f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)$$ for all positive $x,y$,…
Daniel Lee
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How to find a solution for the functional equation $\phi(f(x))=f'(x)\phi(x)$?

Given $f$, how to find some $\phi$ so that $\phi(f(x))=f'(x)\phi(x)$? If you noticed, the given functional equation is of the form of Julia's Equation. However, I cannot seem to find a good article on this equation, so I've come here to ask for…
Sam
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A weird functional equation: $f\big(f(x)\big)=\big(f(x)+1\big)x$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function such that $$f\big(f(x)\big)=\big(f(x)+1\big)x$$ for all real $x$, $f(-1)=0$ and $f(0)=-1$. Find all such functions $f$.
John
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Find all functions $f(x)$ such that $f\left(x^2+f(y)\right)$=$y+(f(x))^2$

Let $\mathbb R$ denote the set of all real numbers. Find all function $f: R\to \ R$ such that $$f\left(x^2+f(y)\right)=y+(f(x))^2$$ It is the problem. I tried to it by putting many at the place of $x$ and $y$ but I can't proceed. Please somebody…
Sufaid Saleel
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If $ f \bigl ( x - f ( y ) \bigr ) = f ( - x ) + \bigl ( f ( y ) - 2 x \bigr ) \cdot f ( - y ) $ what is $ f ( x ) $

Determine all functions $ f : \mathbb R \to \mathbb R $ such that $$ f \bigl ( x - f ( y ) \bigr ) = f ( - x ) + \bigl ( f ( y ) - 2 x \bigr ) \cdot f ( - y ) $$ for all $ x , y \in \mathbb R $. It's easy to see that $ f ( x ) = x ^ 2 $ is a…
user302454
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