Question

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

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

Answer 1

Answer: (C)

$\frac{1-2^{-10}}{3^{10}-2^{-10}}$

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

$g\left(x\right)=3^{10}x-1$

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

$\Rightarrow x(6^{10}-1)=2^{10}-1$

$\Rightarrow x=\frac{2^{10}-1}{6^{10}-1}$

$\Rightarrow x=\frac{1-2^{-10}}{3^{10}-2^{-10}}$