If f ( x ) + 2 f ( 1 x ) = 3 x , x 0 and S = { x : f ( x ) = f ( - x ) } ; then S: [2016]

Question

If $f (x) + 2 f (\frac{1}{x}) = 3 x, x \neq 0$ and

$S = {x \in ℝ : f (x) = f (- x)};$ then S: [2016]

Smart Achievers · Accepted Answer

(1)

$f (x) + 2 f (\frac{1}{x}) = 3 x \dots (i)$

$Replace x by \frac{1}{x}, f (\frac{1}{x}) + 2 f (x) = \frac{3}{x} \dots (i i)$

$Subtract (i) from 2 \times (i i) :$

$2 f (\frac{1}{x}) + 4 f (x) = \frac{6}{x}$

$\underline{f (x)} + \underline{2 f} (\frac{1}{x}) = \underline{3 x}$

__________________ $\Rightarrow f (x) = \frac{2}{x} - x$

$\Rightarrow 3 f (x) = \frac{6}{x} - 3 x$

$f (x) = f (- x) \Rightarrow \frac{2}{x} - x = - \frac{2}{x} + x \Rightarrow x = \frac{2}{x}$

$\Rightarrow x^{2} = 2$ $\Rightarrow$ $\Rightarrow x = \sqrt{2}, - \sqrt{2}$

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