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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Strictly Increasing Function $f$ Satisfying $f(f(x))=3x$

Find all strictly increasing functions that satisfy $$f(f(x))=3x\text.$$ Obviously $\sqrt{3} x$ is an answer, but I am having a hard time proving that is the only solution. Any ideas or solutions? Thanks for the help!
Baker5680
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Differential Equation with inverse function $\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$

$$\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$$ I know $f(x) = ax+b$ is a solution. How can I find other solutions?
ftor
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Determining all functions satisfying the functional equation $ f \big( x + y f ( x ) \big) + f ( x y ) = f ( x ) + f ( 2019 y ) $

I came up with the following functional equations problem, which is from Romanian Master of Mathematics 2019 Day 2, Problem 5 (still lack of solution by 2019 February 28th). I found this problem interesting, but I do not know how to do it. I want to…
Hamio Jiang
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How to find $ f(x)$ if $f(1-f(x))=x$ for all $x$ $\in \mathbb{R}$

How can I determine $ f(x)$ if $f(1-f(x))=x$ for all real $x$? I have already recognized one problem caused from this: it follows that $ f(f(x))=1-x $, which is discontinuous. So how can I construct a function $f(x)$? Best regards and thanks, John
John Doe
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$f(x^2+yf(x))=xf(x+y)$.Find f

Find all real functions $f:R\mapsto R$ satisfying the relation $$f(x^2+yf(x))=xf(x+y).$$ My Answer: Putting $y=0$ we get, $f(x^2)=xf(x)$. o Which implies $\frac{f(x^2)}{x^2}=\frac{f(x)}{x}$. Let $g(x)=\frac{f(x)}{x}$.(Assume $x\ne 0$) Hence,…
Sufaid Saleel
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Solve Functional Equation $3f(x)=f\left(\frac{x}{3}\right)+f\left(\frac{1+x}{3}\right)$

Is there any general strategy for the (non-trivial) solutions to equations similar to $$ 3f(x)=f\left(\frac{x}{3}\right)+f\left(\frac{1+x}{3}\right)? $$ with $x \in \mathbb{R}$. I can…
Benedict W. J. Irwin
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Function Equation Question Involving $f(xy)=f(x)f(y)-f(x+y)+1$

Let $\mathbb{Q}$ denote the set of rational numbers. Find all functions from $\mathbb{Q}$ to $\mathbb{Q}$ which satisfy $f(1)=2$; and $f(xy)=f(x)f(y)-f(x+y)+1$. I know that $f(x)=x+1$ satisfies the equation but I don't know how to prove if this…
user561860
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Functional equation: $ f(x^2+x+3)+2f(x^2-3x+5)= 6x^2 -10 x + 17$

$f: \mathbb R \to \mathbb R, f(x^2+x+3)+2f(x^2-3x+5)= 6x^2 -10 x + 17 \forall x \in \mathbb R $, then find the function $f(x)$ I have: $f(15/4)= 16/3$ (both quadratics intersect at $x=1/2$) $f(3)= 3$ (by substituting $x=0$ and $x=1$ and then…
Archer
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Concerning the nonlinear functional equation $f(x)f(x) =x+1+f(x+1)$

There's a problem I've been working on for awhile that involves some hefty functional equations. For example, I may have something along the lines of $$ f(x)f(x) =x+1+f(x+1) $$ I've tried several different methods of attack (the farthest I ever got…
Mr. G
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Do the solutions to functional equation $f(f(z))=f(z-c)+c$ necessarily have a fixed point?

Let a function $f(z)$ defined for all $z\in\mathbb{Z}$ (without any further restrictions) satisfy the equation: $$ f(f(z))=f(z-c)+c,\tag{1} $$ where $c\ne0$ is an integer constant. Is the following statement valid: There exists $z_0$ such…
user
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Can we assume $ f ( y ) = 0 $ in solving the functional equation $ f \big( x - f ( y ) \big) = f \big( f ( y ) \big) + x f ( y ) + f ( x ) - 1 $?

Here's a problem from one of my books. I believe it's not really a compete solution. Problem. Determine all functions $ f : \mathbb R \to \mathbb R $ such that $$ f \big( x - f ( y ) \big) = f \big( f ( y ) \big) + x f ( y ) + f ( x ) - 1 \tag i…
Mathejunior
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Uniqueness of solution of functional equation

I have a function $f(x_1,x_2) \colon \mathbb{R}^2_{+} \to \mathbb{R}_{+}$ positive homogenous: $$ f(\lambda x_1, \lambda x_2) = \lambda f(x_1,x_2), \; \forall \lambda > 0 $$ and such that $f(x_1,x_2)$ permits decomposition $$ f(x_1,x_2) =…
Appliqué
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Solutions to $(x+y)f(x+y)=xf(x)+yf(y)$

If $x$ is a differentiable function of $t$ and if we define $$ f(x)=\frac{x^\prime}{x} $$ then $f$ satisfies logarithmic-like properties $f(xy)=f(x)+f(y)$ $f(x/y)=f(x)-f(y)$ $f(x^n)=nf(x)$ but $f$ also satisfies the non-logarithmic-like…
John Wayland Bales
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How to solve the functional equation $f(2x)=2f(x)/(1+(f(x))^2)$

solve the functional equation $f(2x)=\dfrac{2f(x)}{1+(f(x))^2}$ I've already known that $\tanh(2x)=2\tanh(x)/(1+(\tanh(x))^2)$. hyperbolic tangent
Tongho
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Functional equations in the form $f(x)-f(x-c)=P_n(x)$

Consider $f:\mathbb R\to\mathbb R$, and let $P_n$ be a polynomial with degree $n\ge 1$. Now, if $$f(x)-f(x-c)=P_n(x)$$ for some fixed constant $c$, then what assumptions on $f$ could let us ensure that $f$ is also a polynomial? This question…
BAI
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