MCQ
જો $f(x) = \log \left[ {\frac{{1 + x}}{{1 - x}}} \right]$, તો $f\left[ {\frac{{2x}}{{1 + {x^2}}}} \right] =$
- A${[f(x)]^2}$
- B${[f(x)]^3}$
- ✓$2f(x)$
- D$3f(x)$
$\therefore \,\,\,f\left( {\frac{{2x}}{{1 + {x^2}}}} \right) = \log \,\left[ {\frac{{1 + \frac{{2x}}{{1 + {x^2}}}}}{{1 - \frac{{2x}}{{1 + {x^2}}}}}} \right] $
$= \log \,\left[ {\frac{{{x^2} + 1 + 2x}}{{{x^2} + 1 - 2x}}} \right]$
$ = \log \,{\left[ {\frac{{1 + x}}{{1 - x}}} \right]^2} = 2\,\log \,\left[ {\frac{{1 + x}}{{1 - x}}} \right] = 2\,f(x)$.
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