If f(x)= ( 1+x 1-x ), then f ( 2x 1+x^2 ) is equal to: - Vidyadip

MCQ

If $\text{f(x)}=\log\Big(\frac{1+\text{x}}{1-\text{x}}\Big),$ then $\text{f}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ is equal to:

A
$\{f(x)\}^2$
B
$\{f(x)\}^3$
✓
$2 \mathrm{f}(\mathrm{x})$
D
$3 f(x)$

✓

Answer

Correct option: C.

$2 \mathrm{f}(\mathrm{x})$

$2 \mathrm{f}(\mathrm{x})$

Solution:
$\text{f(x)}=\log\Big(\frac{1+\text{x}}{1-\text{x}}\Big)$
Then, $\text{f}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)=\log\Bigg(\frac{1+\frac{2\text{x}}{1+\text{x}^2}}{1-\frac{2\text{x}}{1+\text{x}^2}}\Bigg)$
$=\log\Bigg(\frac{\frac{1+\text{x}^2+2\text{x}}{1+\text{x}^2}}{1-\frac{2\text{x}}{1+\text{x}^2}}\Bigg)$
$=\log\bigg(\frac{(1+\text{x})^2}{(1-\text{x})^2}\bigg)$
$=2\log\Big(\frac{1+\text{x}}{1-\text{x}}\Big)$
$=2(\text{f(x)})$

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