$I f f (x) i s a p o l y n o m i a l f u n c t i o n s u c h t h a t f (x) f (\frac{1}{x}) = f (x) + f (\frac{1}{x}) a n d f (3) = - 80 t h e n f (x) - f (\frac{1}{x}) =$

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$x^{4} + \frac{1}{x^{4}}$

$\frac{1}{x^{4}} - x^{4}$

$x^{4} - \frac{1}{x^{4}}$

$- \frac{1}{x^{4}} - x^{4}$

answer is C.

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Detailed Solution

$\begin{array}{l} I f f (x) . f (\frac{1}{x}) = f (x) + f (\frac{1}{x}) t h e n f (x) = 1 \pm x^{n} \\ G i v e n f (3) = - 80 \Rightarrow f (x) = 1 - x^{n} \Rightarrow 1 - 3^{n} = - 80 \\ \Rightarrow 3^{n} = 81 \Rightarrow 3^{n} = 3^{4} \Rightarrow n = 4 \\ f (x) = 1 - x^{4} \Rightarrow f (x) - f (\frac{1}{x}) = 1 - x^{4} - (1 - \frac{1}{x^{4}}) = - x^{4} + \frac{1}{x^{4}} \end{array}$