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 the value of K in a specific case of a cartesian plane

I have a linear equation of a line in a cartesian plane $r:= \{(x,y) \in \mathbb{R}^2 \mid kx-(k+1)y+k-1=0, \,\, k \in \mathbb{R}\}$ and I have to find the value of k so that the line intersects the x axis in a x positive point. Any tips?
Kevin
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A functional equation that is equal to 7x

I wish to find all of the functions $f:\mathbb R \to \mathbb R$ such that $$ f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x $$ for all nonzero $x$. I have tried plugging in $\frac{x-1}{x}$, but that has been of no avail
bob the pie
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solve functional equation: $[f(x)]^2-[f(y)]^2$=$f(x+y)f(x-y)$

i am trying to solve following problems and please guys help me suppose that,there is given following equation $[f(x)]^2-[f(y)]^2$=$f(x+y) \cdot f(x-y)$ there was said that,it requires some knowledge of calculus,first of all i factor this equation…
dato datuashvili
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Solving an equation for a function: $1-\frac{f(x)}{f(ax)} = (1-a)^2x^2$

I am trying to do some proof and in connection with that this question arose: Can you find a decreasing function so that $$ 1-\frac{f(x)}{f(ax)} = (1-a)^2x^2 $$ where $0\leq a \leq 1$ and $x$ is positive? I have tried to plug in various guesses for…
user126540
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Functional equation $f(x)=f(x-1)+f(x-a)$

Please help to solve functional equation for real numbers We have: $a\in \mathbb{R}$ and $a>1$ $f_a(x)=1$ if $x
norman
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Trigonometric functional equation $g(x)f(y) = f \left(\frac{x + y}{2}\right)^2-f\left(\frac{x-y}{2}\right)^2$

Find all functions $ f,g : \mathbb R \to \mathbb R $ that satisfy the functional equation $$g(x)f(y) = f \left(\frac{x + y}{2}\right)^2-f\left(\frac{x-y}{2}\right)^2$$ for all $x,y \in \mathbb R$. I need hints for this problem.
M'smary
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Solving the functional equation $f(x)=f(f(x-p))+q$

I can see that $f(x) = x + (p-q)$ is a solution. Is this the only possible solution?
Vaidhy
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Show that f is linear

Let $f : \mathbb R \to \mathbb R$ be a solution of the additive Cauchy functional equation satisfying the condition $$f(x) = x^2 f(1/x)\quad \forall x \in \mathbb R\setminus \{0\}.$$ Then show that $f(x) = cx,$ where $c$ is an arbitrary constant.
user217960
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The functional equation $f(x + y) = f(x) + f(y) + xy$

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the functional equation $$f(x + y) = f(x) + f(y) + xy\text,$$ for all $x,y \in \mathbb{R}$. I need hints for this problem.
Nikolai
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Given that $f(x^2)=f(x)^2$ and $f(x+1)=f(x)+1$, try to find $f$

Possible Duplicate: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that for all reals $x$ $f(x^2)=f(x)^2$ and $f(x+1)=f(x)+1$ Can we show that $f(x)=x$ for all reals $x$? Do we need additional assumptions…
Zero
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Functional equation $f(x+1)-f(x)=nf(x)$

I want to know how to solve this problem on functions. It seems that it may probably involve discrete differentiating. Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying $$f(x+1)-f(x)=nf(x)\text,$$ where $\mathbb{R}$ is the set of…
Andy Lee
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Functions such that $f(f(n))=n+2015$

Is there a function $f:\mathbb N \to \mathbb N$ such that $\forall n \in \mathbb N, f(f(n))=n+2015$ ? Here's what I've done: Assuming such a function exists,…
Gabriel Romon
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Additive & Multiplicative Link

Is there an isolative property about the following multivariable equation: $$f_1(x)g_1(y)=f_2(x)+g_2(y)$$ That is, is this equation rearrangable such that it may be put in the form $\Omega_1(x)=\Omega_2(y)$ where…
Keith Afas
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functional equation for $x^2$ $f(f(x))=x^4$

If $f(f(x))=x^4$ for all real $x$ and $f(1)=1$ find $f(0)$. It seems that $f(x)=x^2$ but can we solve without this explicit form of $f$?
Petros
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Cauchy's functional equation with polynomial: $ f ( u + a v ) = f ( u ) + f ( a v ) + P _ n ( u , v ) $

Let $ f : \mathbb R \to \mathbb R $ be a continuous function that satisfies $$ f ( u + a v ) = f ( u ) + f ( a v ) + P _ n ( u , v ) $$ where $ a $ is a known constant and $ P _ n ( u , v ) $ is a polynomial in $ u $ and $ v $ of degree $ n $. Is $…
user103828
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