In algebra, one of the most important concepts is Solving functional equations.

To find exact solution of functional equation above relations alone are not sufficient we need more information about the functions. Solution of functional Equation.

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Find all functions f: R+!R+ such that for all positive real numbers x;y, we have: f(x+ f(y)) = yf(xy+ 1): Solution. First, we apply Too Deli by trying to force f(x+f(y)) and f(xy+1) obtain the same value.

A classic example of such a function is because . Cyclic functions can significantly help in solving functional identities. Consider this problem: Find such that . In this functional equation, let and

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functional equation. Problem 3. The function f : R→R satisﬁes x + f(x)= f(f(x)) for every x ∈ R. Find all solutions of the equation f(f(x))=0. Solution. The domain of this function is R, so there

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FUNCTIONAL EQUATIONS 7 Exercise 5.2 Solve the equation f(x + y) = f(x)f(y) where x;y are any real numbers, where f is continuous/bounded. Exercise 5.3 Solve the equation f(x + y) = f(x) +