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 all possible values for $f(2018)$ if $f(x)\cdot f(y)=f(x-y)$

Let $f: \mathbb R \to \mathbb R$ be a function such as $$f(x)\cdot f(y)=f(x-y).$$ Find all possible values for $f(2018)$. All I got is that $f(x)=0$ or $f(0)=1$ (when I put $y=0$) and $f^2(x)=f^2(y)$ (when I put $x=y$).
TheRlee
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Combine two equation

I have two equations with this format: $$Ds= A+A^2+\alpha_1\tag{1}$$ and $$Ds= M+M^2+\alpha_2 \tag{2}$$ Knowing that $(1)$ explains 72% of $Ds$ and $(2)$ 20%. I want to combine these two equations into one and know how much this equation explains.…
Sosa Sebastian
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1 answer

how to construct this function using identities

Given that $f(\sin x)+f(\cos x)= \tan x$. My attempt to find $f(x)$ is by guessing terms to meet the condition. Please help. Note $0 < x < \frac{\pi}{2}$.
pan dong
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2 answers

How to solve functional equation?

I have been struggling to solve this functional equation. Could anyone suggest ways to solve it? It's this: $$f\left(\frac{x-3}{2x+4}\right)=\frac{x+1}{3x-1}.$$ How do I solve it? Thank you in advance!
McLinux
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Find a function which satisfies the relation $f(x+1)=(f(x)+1)^{1/2}$

The original question was to find the value this function approaches as $x$ goes to infinity given that the limit exists. This is easy to figure and turns out to be $(1+5^{1/2})/2$. I was though, thinking, if it were possible to come up with such a…
user3611230
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2 answers

Find all function such that $f(x)-f(y) = (x -y)g(\sqrt{xy})$

Find all functions $f, g$ that satisfy the functional equation $$ f(x)-f(y)= (x -y)g(\sqrt{xy}) \quad \forall\ x,y>0. $$
lavinia
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$\boxed{1} f(x)=\frac{2-x^2}{2}f(\frac{2-x^2}{2})$

Let $f(x)$ be a continuous function from $[-1,1] \rightarrow \mathbb{R}$ having the following properties: $\boxed{1} f(x)=\dfrac{2-x^2}{2}\cdot f\!\left(\dfrac{2-x^2}{2}\right)$ $\boxed{2} f(0)=1$ $\boxed{3}…
user321656
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1 answer

Prove that $f(x) = x$ is the unique solution to the following functional equation

Let $f : [0 , \infty) \to \mathbb{R},$ continuous at $ x_0 = 0$ satisfying $$f(3x) - 2x = f(x)$$ Prove that $f(x) = x$.
Eduard6421
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inverse function , asymptotics ..

let be a function given by $ f^{-1} (x) = \sqrt x + G(x) $ with $ G(x)= \sum_{n=0}^{N}a_{n} \sin(nx+ \pi/4) $ finite fourier series with N big my questio is , if for $ x \rightarrow \infty$ the function can be asymptotically defined by $ f(x) \sim…
Jose Garcia
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1 answer

$x^3[f(x+1)-f(x)]=1$

Given that f is continuous and $x^3[f(x+1)-f(x)]=1$, determine $\lim_{x\rightarrow \infty}f(x)$.
Benji
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1 answer

$x^3[f(x+1)-f(x-1)]=1$

Given that $x^3[f(x+1)-f(x-1)]=1$, determine $\lim_{x\rightarrow \infty}(f(x)-f(x-1))$ explicitly.
Benji
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fancy about $f(x+a)=f(x)$ , where $a$ is any non-real complex number

It is well-known that when $a$ is any non-zero real number, the most general solution of $f(x+a)=f(x)$ should be $f(x)=\Theta(x)$, where $\Theta(x)$ is an arbitrary periodic function with period $|a|$ . Now when $a$ is a non-real complex number,…
doraemonpaul
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Functional equaliton on continuous function $f:[0,1]\to \mathbb{R}$

Let $f:[0,1]\to \mathbb{R}$ be continuous, and suppose that $f(0)=f(1).$ Show that there is a value $x\in [0, 1998/1999]$ satisfying $f(x)=f(x+1/1999)$.
Raheem Najib
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Fredholm-like equation

I have the following equation: $$\lambda=\displaystyle\int_{a}^b f(x)g(x)dx$$ Where $\lambda$ is a constant and I know the expresión for f(x). Is there any way of extracting the fucntion g(x)? I have been looking at Fredholm equations, but you need…
DanielPerez
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3 answers

Find all real functions that satisfy the functional equations $f(x+y) = f(x) + f(y)$ and $f(xy)=f(x)\,f(y)$

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the two functional equations $f(x + y) =f(x)+f(y)$ and $f(xy)=f(x)\,f(y)$.
mikee
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