Question

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

The correct option is D $\frac{1-2^{-10}}{3^{10}-2^{-10}}$

Given:$f\left(x\right)=2^{10}x+1$ and $g\left(x\right)=3^{10}x-1$

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

$\Rightarrow f(3^{10}x-1)=x$

$\Rightarrow 2^{10}(3^{10}x-1)+1=x$

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

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

$\Rightarrow -2^{10}+1=x(1-2^{10}{3}^{10})$

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

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

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

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

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