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  4. If f(x)= log((1+x/1-x)) and g(x)=(3x+x3/1+3x2), then f(g(x)) is equal to

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Q. 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

Relations and Functions - Part 2

Solution:

$f(g(x))=log\left(\frac{1+g(x)}{1-g(x)}\right)=\log\,\left(\frac{1+\frac{3x+x^3}{1+3x^2}}{1-\frac{3x+x^3}{1+3x^2}}\right)$
$=\log\left(\frac{1+3x^2+3x+x^3}{1+3x^2-3x-x^3}\right)$
$=\log\left(\frac{1+x}{1-x}\right)=3 \log \left(\frac{1+x}{1-x}\right)=3f(x)$.