If f (x) = _ e ((1 - x)/(1 + x)), |x| < 1, then f( 2x 1 + x ^ 2 )...

$f\left(x\right) = \ln \left(\dfrac{1 - x}{1 + x}\right)$

$⇒ f\left(\dfrac{2x}{1 + {x}^{2}}\right) = \ln \left(\dfrac{1 - \dfrac{2x}{1 + {x}^{2}}}{1 + \dfrac{2x}{1 + {x}^{2}}}\right)$

$= \ln \left(\dfrac{1 + {x}^{2} - 2x}{1 + {x}^{2} + 2x}\right)$

$= \ln \left({\left(\dfrac{1 - x}{1 + x}\right)}^{2}\right)$

$= 2\ln \left|\dfrac{1 - x}{1 + x}\right|$

$= 2\ln \left(\dfrac{1 - x}{1 + x}\right)\left(∵ \left|x\right| < 1\right)$

$= 2f\left(x\right)$