Mathematics - Functions Question with Solution | TestHub

$\begin{array}{rl}& f(f(x))=\frac{2f(x)}{\sqrt{1+9f^2(x)}}=\frac{4x}{\sqrt{1+9x^2+9.2^2{x}^2}}\\ & f(f(f(x)))=\frac{2^{3}x/\sqrt{1+9x^2}}{\sqrt{1+9\left(1+2^2\right)\frac{2^2{x}^2}{1+9x^2}}}=\frac{2^{3}x}{\sqrt{1+9x^2\left(1+2^2+2^4\right)}}\end{array}$ $\therefore$ By observation $\begin{array}{rl}& \alpha =1+2^2+2^4+\dots .+2^{18}=1\left(\frac{{\left(2^2\right)}^{10}-1}{2^2-1}\right)=\frac{2^{20}-1}{3}\\ & 3\alpha +1=2^{20}\to \sqrt{3\alpha +1}=2^{10}=1024\end{array}$