A) \[\frac{3x}{\sqrt{1+{{x}^{2}}}}\]
B) \[\frac{x}{\sqrt{1+3{{x}^{2}}}}\]
C) \[\frac{3x}{\sqrt{1+{{x}^{2}}}}\]
D) None of these
Correct Answer: B
Solution :
\[(fofof)\,(x)=(fof)\,(f(x))=(fof)\,\left( \frac{x}{\sqrt{1+{{x}^{2}}}} \right)\] \[=f\,\left[ \frac{\left( \frac{x}{\sqrt{1+{{x}^{2}}}} \right)}{\sqrt{1+\frac{{{x}^{2}}}{1+{{x}^{2}}}}} \right]=f\,\left( \frac{x\sqrt{1+{{x}^{2}}}}{\sqrt{1+{{x}^{2}}}\sqrt{1+2{{x}^{2}}}} \right)\] \[=f\,\left( \frac{x}{\sqrt{1+2{{x}^{2}}}} \right)=\frac{\frac{x}{\sqrt{1+2{{x}^{2}}}}}{\sqrt{\left[ 1+\frac{{{x}^{2}}}{1+2{{x}^{2}}} \right]}}=\frac{x}{\sqrt{1+3{{x}^{2}}}}.\]
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