Find \( f\left(\frac{2x}{1+x^2}\right) \)
Question:
If
\[ f(x)=\log\left(\frac{1+x}{1-x}\right) \]
then
\[ f\left(\frac{2x}{1+x^2}\right) \]
is equal to
(a) \(\{f(x)\}^2\)
(b) \(\{f(x)\}^3\)
(c) \(2f(x)\)
(d) \(3f(x)\)
Solution:
\[ f\left(\frac{2x}{1+x^2}\right) = \log\left( \frac{1+\frac{2x}{1+x^2}} {1-\frac{2x}{1+x^2}} \right) \]
\[ = \log\left( \frac{(1+x)^2}{(1-x)^2} \right) \]
\[ = 2\log\left( \frac{1+x}{1-x} \right) \]
\[ =2f(x) \]
\[ \boxed{\text{Correct Answer: (c)}} \]