A function f satisfies f(x + y) = f(x) + f(y) + 2xy for all numbers x and y, and f(1) = 3. What is f(3)?
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Answer & Explanation
Correct answer
E
Method 1 (build up step by step): f(2) = f(1) + f(1) + 2(1)(1) = 3 + 3 + 2 = 8; then f(3) = f(2) + f(1) + 2(2)(1) = 8 + 3 + 4 = 15. Method 2 (closed form): the rule forces f(n) = n² + 2n for integer n, since f(1) = 1 + 2 = 3 fits; then f(3) = 9 + 6 = 15.
The choices cannot be back-solved because f is defined only implicitly through the functional equation, so each value of f must be constructed from f(1) upward; the non-obvious second move is carrying the 2xy cross-term with the correct arguments (x = 2, y = 1) at the final composition step rather than treating f as additive.
Choice 9 treats f as linear. Choice 11 drops the final cross-term. Choice 13 under-weights it. Choice 14 drops the factor of 2 in f(2).