A) \[f(3x)\]
B) \[{{(f(x))}^{3}}\]
C) \[3f(x)\]
D) \[-f(x)\]
Correct Answer: C
Solution :
\[f(x)=\log \left( \frac{1+x}{1-x} \right)\] and \[g(x)=\frac{3x+{{x}^{3}}}{1+3{{x}^{2}}}\] \[\therefore \] \[f(g(x))=\log \left( \frac{1+g(x)}{1-g(x)} \right)\,\] \[=\log \left( \frac{1+\frac{3x+{{x}^{3}}}{1+3{{x}^{2}}}}{1-\frac{3x+{{x}^{3}}}{1+3{{x}^{2}}}} \right)\] \[=\log \,\left( \frac{1+3{{x}^{2}}+3x+{{x}^{3}}}{1+3{{x}^{2}}-3x-{{x}^{3}}} \right)\] \[=\log \left[ \frac{{{(1+x)}^{3}}}{{{(1-x)}^{3}}} \right]\] \[=3\log \left( \frac{1+x}{1-x} \right)=3f(x)\]
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