# 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).

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### Is there any functional equation $f(ab+cd)= f(a)+f(b)+f(c)+f(d)$?

I am looking for a real, continuous function that satisfies the functional equation $$f(ab+cd)= f(a)+f(b)+f(c)+f(d)$$ where $a,b,c,d$ are real. This is equivalent to a function satisfying these two ...
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### find all possible functions : $f(a)f(b)-6ab=\frac{3}{2}f(a+b)$

I'm trying to find all possible functions that satisfy this functional equation: $f(a)f(b)-6ab=\frac{3}{2}f(a+b),$ $f\in \mathbb{R}.$ My attempt : $a=b=0$ then $f(0)=0$ or $1.$ But I don't ...
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### Functional equations in one variable.. [closed]

How do you solve the functional equation involving only one variable...what if and if not given that $f(x)$ is a polynomial... Say for example $f(x)=f(x-1) +2x$
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### $‎\lim_{n\to\infty} (xf(n) + \sum_{k=1}^n f(k) - f(k+x))‎$‎ is‎ ‎convergent for $x\geq 1$. [closed]

‎‎Let $f:[1‎, +‎\infty)\rightarrow\mathbb{R}$ be a function such that ‎$‎\lim_{n\to\infty} (f(n) - f(n+1)) = 0‎$‎‎. ‎Suppose that $\lambda:[1‎, +‎\infty)\rightarrow\mathbb{R}$‎ ‎is an exp-convex ...
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### IMO 1993 b2 proof

"Let $\mathbb{N}=\{ 1,2,3,\ldots \}$. Determine if there exists a strictly increasing function $f:\mathbb{N}\to\mathbb{N}$ such that 1) $f(1)=2$ 2) $f(f(n))=f(n)+n$ for all $n$." Of the solutions ...
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### Find all functions $f : \mathbb{(0,\infty})\to\mathbb{R}$ such that $f(x+y) = xf(y)+ yf(x)$

Find all functions $f : \mathbb{(0,\infty)}\to\mathbb{R}$ such that $f(x+y) = xf(y)+ yf(x)$ if $f$ is continuous at $x=1$ This problem was looking quite easy at first but the domain of positive ...
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### Solve functional equation $f(z)=c+zf(z^2)$ with series expansion?

Let the functional equation $(1)$ be given as $$f(z)=c+zf(z^2) \tag{1}$$ where $c \in\mathbb R$ and $c \neq 0$. How can this functional equation be solved with series expansion (power, Taylor or ...
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### Find functions $f,g$ such that $f(x+y)=g(x·y)$ for all $x,y$? [closed]

My approach towards this question was that first I putted x=0 and then y=0 which yields f(x)=f(y)and f(x)=g(0), and again for x=1 & y=1 it gives g (x)=f(1+x), and so on. My query is that what ...
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### Solving the equation $f(x)=f^{-1}(x)$. [duplicate]

Exactly under what conditions would the equality $f(x)=f^{-1}(x)$ hold? The proof attached below considers a special case when $f$ is strictly increasing. The theorem then says that the set of ...
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### Difficulties with Inner products and polarization identities

I am discussing the general inner product space. Here is what Polarization Identities mean. I denote the inner product by $(x,y)$. I am having a difficult time with the polarization identities. Of ...
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### Find all real functions such that $(x + 1)f(xf(y)) = xf(y(x + 1))$

Find all real functions of real variable such that $$(x + 1)f(xf(y)) = xf(y(x + 1))$$ Let $a=f(0)$. For $y=0$ we get $(x+1)f(ax) = ax$, so if $a\ne 0$ we get $$f(x) = {ax\over x+a}$$ which is actual ...
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### Functional Equation Problems

If $\Bbb N$ denotes all positive integers. Then find all functions $f: \mathbb{N} \to \mathbb{N}$ which are strictly increasing and such for all positive integers $n$, we have: $$f(f(n)) = n+2$$ So ...
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### Functional equation in single-variable calculus

Suppose we know that $f(x) \in C^2$ and $f(x)$ defined for all real numbers. Furthermore, $f(x)$ has following property: $$\forall x,y \in \Bbb R \quad f(x+y) - f(x) = yf'(x + \frac{y}{2})$$ How to ...
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### Functional Equation $f(x)f(f(x)+\frac{1}{x})=1$

I'd like to ask how to find all solutions to the functional equation $f(x)\cdot f(f(x)+\frac{1}{x})=1$, where $f: (0, +\infty)\to\mathbb{R}$ is strictly increasing?
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### Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R}$ , $f(xf(x)+f(y))=x^2+y$

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y, \in \Bbb{R}$ , $f(xf(x)+f(y))=x^2+y$ We can easily get a strong condition $f(f(y))=y$ by setting $x=0$ . By this equation we ...
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### Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R}$ , $f(f(x)+yz)=x+f(y)f(z)$
Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R}$ , $f(f(x)+yz)=x+f(y)f(z)$ I was told to do this by proving $f$ is injective and surjective. I have proved it this ...