If f(x) + 2= 3x, x 0, and S = {x R : f(x) = f(

Engineering

Mathematics

functional equation

Question

If f(x) + 2 $f (\frac{1}{x})$ = 3x, x $\neq$ 0, and S = {x $\in$ R : f(x) = f(– x)}; then S

is an empty set.

contains more than two elements.

contains exactly two elements.

contains exactly one element.

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

replace x by $\frac{1}{x}$ , we get

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

$- 3 f (x) = 3 x - \frac{6}{x} \Rightarrow f (x) = \frac{2}{x} - x = \frac{2 - x^{2}}{x}$

$∵ f (- x) = f (x) \Rightarrow \frac{2 - x^{2}}{- x} = 2 - x^{2} = 0$

$\Rightarrow x = \pm \sqrt{2} .$