Mathematics - Functions Question with Solution | TestHub

$f\left(x\right)=\mathrm{l}\mathrm{n}\left(\frac{1-x}{1+x}\right)$

$\Rightarrow f\left(\frac{2x}{1+x^2}\right)=\mathrm{l}\mathrm{n}\left(\frac{1-\frac{2x}{1+x^2}}{1+\frac{2x}{1+x^2}}\right)$

$=\mathrm{l}\mathrm{n}\left(\frac{1+x^2-2x}{1+x^2+2x}\right)$

$=\mathrm{l}\mathrm{n}\left({\left(\frac{1-x}{1+x}\right)}^2\right)$

$=2\mathrm{l}\mathrm{n}\left|\frac{1-x}{1+x}\right|$

$=2\mathrm{l}\mathrm{n}\left(\frac{1-x}{1+x}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\because \left|x\right|<1\right)$

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