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Algebra

Functional Equations

Functional Equations - 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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