If f 2(x) · f \left(\right. \frac{1 - x}{1 + x} \left.\right) = x^{3} \forall x \in R – {–1, 1} and f(x) ≠ 0, then which of the following is correct.

Engineering

Mathematics

Piecewise Defined Functions

Question

If f ²(x) · f $(\frac{1 - x}{1 + x}) = x^{3} \forall x \in R$ – {–1, 1} and f(x) ≠ 0, then which of the following is correct.

f(2) · f(–2) = 16

f (2) f(–2) = 8

f (2) = –12

f (3) = –18

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

Replace x by $(\frac{1 - x}{1 + x})$ , we get

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

From $\frac{{(1)}^{2}}{(2)}$ , we get

$f^{3} (x) = x^{6} {(\frac{1 + x}{1 - x})}^{3}$

$∴ f (x) = x^{2} (\frac{1 + x}{1 - x})$

$∴ f (2) = 4 (\frac{3}{- 1}) = - 12$

and f (–2) = 4 $(\frac{- 1}{3}) = \frac{- 4}{3}$
f (2) f (–2) = 16
and f (3) = $\frac{9 \times 4}{- 2}$ = –18

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