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Functional Equations

Functional equations are equations where the unknowns are the functions. Rather than solving for x, you solve for the function in questions like "Find all functions that have these properties."

Problem Solving

         

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: \[\begin{align} & f(1)=1, f(2)=2, \text{ and}\\ & f(n+2)=f(n+1)+\left(f(n)\right)^2+3581 \text{ for } n \geq 1. \end{align} \] 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 \(f(10)? \)

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