Algebra 2 - by Vlad Vasilescu (W)

Algebra Level pending

A function γ : R → R \gamma : \mathbb{R \rightarrow R} has the property that γ ( x − y ) = γ ( x ) + γ ( y ) − 2 x y \gamma(x-y) = \gamma(x) + \gamma(y) - 2xy for all real x x and y y . Find γ ( 3 ) \gamma(3) .

Notation : R \mathbb R is the set of all real numbers.


The answer is 9.

Vlad Vasilescu by Vlad Vasilescu , 22, Romania

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2 solutions

Chew-Seong Cheong
Nov 13, 2016

y = x : γ ( x − x ) = γ ( x ) + γ ( x ) − 2 x 2 γ ( 0 ) = 2 γ ( x ) − 2 x 2 . . . ( 1 ) x = 0 : γ ( 0 ) = 2 γ ( 0 ) ⟹ γ ( 0 ) = 0 ( 1 ) : 2 γ ( x ) − 2 x 2 = 0 ⟹ γ ( x ) = x 2 ⟹ γ ( 3 ) = 3 2 = 9 \begin{aligned} y=x: \quad \gamma(x-x) & = \gamma(x) + \gamma(x) - 2x^2 \\ \gamma(0) & = 2\gamma(x) - 2x^2 & ...(1) \\ x=0: \quad \quad \quad \gamma(0) & = 2\gamma(0) \\ \implies \gamma(0) & = 0 \\ (1): \quad 2\gamma(x) - 2x^2 & = 0 \\ \implies \gamma(x) & = x^2 \\ \implies \gamma(3) & = 3^2 = \boxed{9} \end{aligned}

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Vlad Vasilescu
Nov 13, 2016

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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