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The function f : R -> R satisfies

f(x)f(y) = f(x + y) + f(xy)

for all real numbers x and y.  Find all possible functions f.

 May 14, 2022
 #1
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What if x = y = 0?

 

\((f(0))^2 = f(0) + f(0)\\ \)

Let x = f(0), then x^2 - 2x = 0. This implies f(0) = 0 or f(0) = 2.

 

Case 1: f(0) = 2

Substituting x = 0 gives:

\(f(0) f(y) = f(y) + f(0)\\ f(y) = 2\)

Note that this holds for any real number y. That means the constant function \(f(x) = 2\) is a solution to this functional equation.

 

Case 2: f(0) = 0

Substituting x = 0 gives:

\(f(0) f(y) = f(y) + f(0)\\ f(y) = 0\)

Note that this holds for any real number y. That means the constant function \(f(x) = 0\) is a solution to this functional equation.

 

The only solutions to this functional equation are \(f(x) = 0\) or \(f(x) = 2\).

 May 14, 2022

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