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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For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)\geq 0$, we have: $2f(-a)+f(b)\leq 0$. Then, $f$ must be an odd function?

For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)=0$, we have: $2f(-a)+f(b)=0$. For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)>0$, we have: $2f(-a)+f(b)<0$. Also, for all $a$, there ...
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$f(a)+f(-b)=0$ implies $f(-a)+f(b)=0$. Then $f$ is an odd function?

If $f(a)+f(-b)=0$ for some $(a,b)$, then, $f(-a)+f(b)=0$. Also, for all $a$, there exists $b$ such that $f(a)+f(-b)=0$. $f:\mathbb R\to\mathbb R$ is continuous. Based on the above conditions, can we ...
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Find function $f$ such that $f(\frac{x-3}{x+1})+f(\frac{x+3}{1-x})=x$

I am looking for functions $f:\Bbb R \to \Bbb R$ satisfying $$f\Big(\frac{x-3}{x+1}\Big)+f\Big(\frac{x+3}{1-x}\Big)=x$$ I used the substitution $x=\cos(2t)$ for $x\in (0,2\pi)$, to use the ...
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Find the smallest value at x=25

A function g: is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers m and n. Let f be an adjective function such that the value of $f(1)+f(2)+f(3)+...+f(30)$ is minimized. Find ...
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Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(f(x)+xf(y))=x+yf(x).$

Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(f(x)+xf(y))=x+yf(x).$ My attempt: \begin{align} P(0, y): \; & f(f(0))=yf(0)=0. \\ \Rightarrow \; & f(0)=0. \\ \ \\ P(x, 0): \; & f(f(...
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Find all functions that satisfy $f(x^2f(y)^2)=f(x)^2f(y)$.

Find all $f:\mathbb Q_{>0}\to \mathbb Q_{>0}$ such that $$f(x^2f(y)^2)=f(x)^2f(y)$$ This is from the IMO Shortlist 2018, and I just want to know if my solution is valid, here is the official ...
52 views

$\frac{g(x,y)}{f(x)-f(y)}$ is increasing with $x$ and decreasing with $y$. Then, what is $g$?

Background: $x,y,a$ are real numbers. $a>0$. $f(x), g(x,y), h(x,y)$ are continuous real functions. $g,h$ are defined on $[0,1]^2$. $$h(x,y)=\frac{g(x,y)}{f(x)-f(y)}>0, \text{when} \ x\neq y.$$ ...