Let f(x) = 2 ^ 10 x + 1 and g(x) = 3 ^ 10 x - 1 . If (fog)(x) = x,...

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

Where,$f\left(x\right) = {2}^{10}x + 1$ ,$g\left(x\right) = {3}^{10}x - 1 .$

$⇒ f\left({3}^{10}x - 1\right) = {2}^{10}\left({3}^{10}x - 1\right) +1$

$⇒ {2}^{10}\left({3}^{10}x - 1\right) + 1 = x$

$⇒ x\left({6}^{10} - 1\right) = {2}^{10} - 1$

$⇒ x = \dfrac{{2}^{10} - 1}{{6}^{10} - 1} = \dfrac{\dfrac{{2}^{10} - 1}{{2}^{10}}}{\dfrac{( 2 \times 3{)}^{10} - 1}{{2}^{10}}}$

$= \dfrac{1 - {2}^{- 10}}{{3}^{10} - {2}^{- 10}}$