Functional Equations

The function $f$ satisfies $(x-y)f(x+y)-(x+y)f(x-y)+4xy(x^2-y^2)=0,$

whenever $|x| \neq |y |$ .

Given that $f(1)=2,$ what is $f(x)$ ?

Function $f(n)$ defined over all positive integers $n$ satisfies the following: $\begin{aligned} & f(1)=1, f(2)=2, \text{ and}\\ & f(n+2)=f(n+1)+\left(f(n)\right)^2+3581 \text{ for } n \geq 1. \end{aligned}$ How many of the $3581$ integers $f(1), f(2), \ldots, f(3581)$ are multiples of $7?$

What is the polynomial $p(x)$ that satisfies $p(x^2+4)=\left(p(x)\right)^2+4 \text{ and } p(0)=0?$

A function $f$ from the positive integers to the positive integers satisfies the conditions
1) $f(n) = 2 f(n-1) + 3$ ,
2) $f(1) = 9 .$

What is the value of $f(9)$ ?

If $f(x)$ is a polynomial satisfying $27 f(x^3) -4f(x^2) - x^6 f(3x) + 46 = 0$ , what is $f(10)?$

Concept Quizzes

Challenge Quizzes

Functional Equations - Problem Solving

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.