f ( x ) f(x) is everywhere!!!!!

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f ( x ) f(x) is a function defined for all real values of x x different from 0 0 . f ( x ) f(x) satisfies: 2 f ( x ) − 3 f ( 1 x ) = x 2 2f(x)-3f(\frac{1}{x})=x^2 .If − f ( 3 ) = a b -f(\sqrt{3})=\frac{a}{b} where a a and b b are prime positive integers find a + b a+b .


The answer is 12.

Lorenc Bushi by Lorenc Bushi , 21, Albania

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1 solution

2 f ( 3 ) − 3 f ( 1 3 ) = 3 2f(\sqrt{3})-3f \left (\frac{1}{\sqrt{3}} \right ) =3 − 3 f ( 3 ) + 2 f ( 1 3 ) = 1 3 -3f(\sqrt{3})+2f \left (\frac{1}{\sqrt{3}} \right )=\frac{1}{3} Applying Cramer’s Rule: \text{Applying Cramer's Rule:} ∣ 2 − 3 − 3 2 ∣ = − 5 \begin{vmatrix} 2 & -3 \\ -3 & 2 \end{vmatrix} = -5 ∣ 3 − 3 1 3 2 ∣ = 7 \begin{vmatrix} 3 & -3 \\ \frac{1}{3} & 2 \end{vmatrix} = 7 f ( 3 ) = − 7 5 f(\sqrt{3}) = -\frac{7}{5} a + b = 12. \boxed{a+b=12.}

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I don't understand matrices...

Bogdan Simeonov - 7 years, 5 months ago

Bogdan, matrices are not the only way to solve this problem...

Lorenc Bushi - 7 years, 5 months ago

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