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  4. Given f (x)=log((1+x/1-x)) and g(x)=(3x+x3/1+3x2), then fog(x) equals

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Q. Given $f \left(x\right)=log\left(\frac{1+x}{1-x}\right)$ and $g\left(x\right)=\frac{3x+x^{3}}{1+3x^{2}}$, then $fog(x)$ equals

Relations and Functions - Part 2

Solution:

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