A) \[{{[f(x)]}^{3}}\]
B) \[{{[f(x)]}^{2}}\]
C) \[-f(x)\]
D) \[f(x)\]
E) \[3f(x)\]
Correct Answer: D
Solution :
Given, \[f(x)=\log \left( \frac{1+x}{1-x} \right)\] \[\therefore \] \[f\left( \frac{3x+{{x}^{3}}}{1+3{{x}^{2}}} \right)-f\left( \frac{2x}{1+{{x}^{2}}} \right)\] \[=\log \left( \frac{1+\left( \frac{3x+{{x}^{3}}}{1+3x2} \right)}{1-\left( \frac{3x+{{x}^{3}}}{1+3{{x}^{2}}} \right)} \right)-\log \left( \frac{1+\frac{2x}{1+{{x}^{2}}}}{1-\frac{2x}{1+{{x}^{2}}}} \right)\] \[=\log {{\left( \frac{1+x}{1-x} \right)}^{3}}-\log {{\left( \frac{1+x}{1-x} \right)}^{2}}\] \[=\log \left( \frac{1+x}{1-x} \right)\] \[=f(x)\]
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