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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solve equations - find a short piece of a wire

Problem: A piece of wire 20 feet long is cut into two pieces so that the sum of the squares of the lengths of the two pieces is 202 square feet. What is the length, in feet, of the shorter piece of wire? This problem looks pretty simple, but while…
learning
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Alternate method for finding $f(x)$

Let f be a real valued function $f:(0,\infty)\to(0,\infty)$ such that it satisfies the relation: $$f(xf(y))=x^2\cdot y^a$$ where $a\in\mathfrak{R}$ then find $f(x)$ and the possible values of $a$. My attempt: Looking at the given relation, I can…
Saransh Kumar
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Number of distinct values

Question: How many possible values of (a, b, c, d), with a, b, c, d real, are there such that abc = d, bcd = a, cda = b and dab = c? I tried multiplying all the four equations to get: $$(abcd)^2 = 1$$ Not sure how to proceed on from here. Won't…
Gummy bears
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Find all functions $f$ over reals such that $f(xy) \geq f(x+y)$ for all $x,y \in \mathbb{R}$.

Find all functions $f$ over reals such that $f(xy) \geq f(x+y)$ for all $x,y \in \mathbb{R}$. We have that $f(x) \geq f(x+1)$ and $f(0) \geq f(1)$. I am wondering how to use these conditions to solve the problem. It seems like using a telescoping…
user19405892
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If $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$, find $f$

Assume $f: (0, \infty) \to \mathbb{R}$ is a continuous function such that for any $x,y > 0$, $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$. Find $f$. I would work with each condition separately then combine later. So for $f(\frac{x+y}{2}) =…
Puzzled417
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Find all functions $f$, defined over real numbers that satisfy $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$

Find all functions $f$, defined over real numbers that satisfy $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$. We can set $x = 0$ and $y = n$ we get $f(n) = f(0)+f(n) \implies f(0) = 0$. Then what I thought of doing was saying $f(x+y)^2 =…
user19405892
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Let $f$ be a function from $Q$ to $Q$ satisfying $f(x+f(y))=f(x).f(y)$. Prove that $f$ is constant

Let $f$ be a function from $Q$ to $Q$ satisfying $f(x+f(y))=f(x).f(y)$. Prove that $f$ is constant I was trying to divide into 3 cases: when f(x) has a root, when f(x)>0 and when f(x)<0 .. But I am having trouble with $f(x)<0$
user321656
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Resolution of equation such that $f(...f(x)...)=x$

I am wondering if it exits a way to find "easely" the solutions of an equation about a function $f$ such that $$f^{(n)}(x)=x$$ where $f^{(n)}$ is the n-th composition of $f$ itself. Obviously the identity is a trivial solution, I'm asking for all…
ParaH2
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Solving $f(2011)=2012$, $f(4xy)=2yf(x+y)+f(x-y)$

How to find the all functions $f$ :$ \mathbb{R}\longrightarrow\mathbb{R}$ such that $f(2011)=2012$,for every $x,y\in\mathbb{R}$ then: $$f(4xy)=2yf(x+y)+f(x-y)$$
Frank
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Does there exist such a function $f(x)$ that $f^n(x)=\left (1-\frac {1}{\sqrt[n]{x}} \right)^n?$

Let $n=\underbrace{11\dots1}_{1996\text{ figures}}$. Does there exist such a function $f(x)$ that for all real $x \ne 0, x \ne 1$ holds $$\underbrace{f \bigg ( f \Big( \dots \big( f}_{n\text{ iteratoins}}(x) \big) \Big) \bigg)=\left (1-\frac…
Roman83
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Functions $f$ which satisfy $f(x^4+2x^2+2)-1=(f(x)-1)^4+4x(f(x)-1)^2+4x^2$

I want to find all functions $f:\mathbb R\to \mathbb R$ which satisfy $$f(x^4+2x^2+2)-1=(f(x)-1)^4+4x(f(x)-1)^2+4x^2$$ for all $x$. I know the solution: the only solution is $f(x)=x$. How can I prove it?
piteer
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If $f(x+y)-f(x-y)=4xy$, then is $f(0)=0$ or $f(0)=1$?

As in the title. It may be very simple, but I'm having difficulty finding the proper substitution.
user263286
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Express $y$ from $\ln(x)+3\ln(y) = y$

i am finding a inverse function to $$y=\frac{e^\frac{x^3}{3}}{x^3}$$ First step: $$x=\frac{e^\frac{y^3}{3}}{y^3}$$ Next: ... $$(xy^3)^3 = e^{{y}^3}$$ and now in this step i have no idea how can i express "y" $$\ln(x)+3\ln(y) = y$$
DavidM
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Determine formula to calculate inverse range between two numbers

I've got a measurement for a graphics processing filter that increases intensity as the value decreases. For whatever reason, this filter's default value (minimum) is 1.75, and I've set the maximum value to 0.55. Because all the other filters in…
brandonscript
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Functional equation $f(h(y)x+y)=g(y)f(x)$

(Note: this is a simplified version of my previous question, which was not answered). I am seeking the solution for the functional equation $f(h(y)x+y)=g(y)f(x)$ where $f,g,h$ are continuous. Clearly, $f$ constant and $g=1$ is a solution. If…
mike
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