Question

Let $f (x)=2^{10}⋅x+1$ and $g(x)=3^{10} ⋅x-1$. If $(fog)(x)=x$, then $x$ is equal to :

• A. $\frac{3^{10} - 1 }{3^{10} - 2^{-10}}$
• B. $\frac{2^{10} - 1 }{2^{10} - 3^{-10}}$
• C. $\frac{1 - 3^{10} }{2^{10} - 3^{-10}}$
• D. $\frac{1 - 2^{10} }{3^{10} - 2^{-10}}$

Answer 1

Answer: (D)

f(g(x)) = x
$f\left(3^{10}x - 1\right) = 2^{10}\left(3^{10}.x - 1\right) = x$
$= \frac{1}{3^{10}-2^{-110}}$
$2^{10}\left(3^{10}x -1\right) + 1 = x$
$x\left(6^{10} -1\right) = 2^{10 }- 1$
$x = \frac{2^{10}-1}{6^{10}-1}\quad\quad=\frac{1-2^{-10}}{3^{10}-2^{-10}}$