Question

If $f\left(x\right)=\frac{x}{\sqrt{1-x^2}},g\left(x\right)=\frac{x}{\sqrt{1+x^2}}$, find $\text{fog}\left(x\right)$.

Answer 1

Given, $f\left(x\right)=\frac{x}{\sqrt{1-x^2}},g\left(x\right)=\frac{x}{\sqrt{1+x^2}}$

Therefore, $f\left(g\right(x\left)\right)=\frac{g\left(x\right)}{\sqrt{1-{\left\{g\right(x\left)\right\}}^2}}$

$\Rightarrow f\left(g\right(x\left)\right)=\frac{\frac{x}{\sqrt{1+x^2}}}{\sqrt{1-{\left(\frac{x}{\sqrt{1+x^2}}\right)}^2}}\Rightarrow f\left(g\right(x\left)\right)=\frac{\frac{x}{\sqrt{1+x^2}}}{\sqrt{\frac{1+x^2-x^2}{1+x^2}}}$

$\Rightarrow f\left(g\right(x\left)\right)=x$