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