A Nice Homemade Functional Equation

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PLAYLISTS 🎵 :

[theelk]

if you do x=x and y=0 you get the answer immediately

[wolliwolfsen]

Why so complicated? The solution is obvious!

[honestadministrator]

at y = 0, f ( x) = 2 x
This satisfies the given expression

[mystychief]

It looks linear. Fill in f(x) = ax+b (twice) then b must be zero and a must be 2.

[iabervon]

If it's true for all y, it's true for y=0, and f(x)=2x. But it's worth checking that this also gives consistent results for non-zero y, in case y the answer is that no function works. However, 2(x+y(2x))=2x+4xy, as needed.

[bkkboy-cmeb]

y=0 → f(x) = 2x
x+yf(x) = x + 2xy
f(x+yf(x)) = f(x + 2xy) = 2x +4xy
∴f(x) = 2x