2019 ACJC P1 Q9 (b)

The function \(\mathrm{g}\) is defined for \(x>0\) as

\(\mathrm{g}:x\mapsto {{2}^{n}}x-1,\frac{1}{{{2}^{n}}}\le x<\frac{1}{{{2}^{n-1}}}\), where \(n\in \mathbb{Z}\).

1. Fill in the blanks.
   
   Hence sketch the graph \(y=\mathrm{g}\left( x \right)\) for \(\frac{1}{4}\le x<1\).[3]
2. Show that \(\mathrm{g}\left( x \right)=\mathrm{g}\left( \frac{x}{2} \right)\).[2]
3. Find the number of solutions of \(\mathrm{g}\left( x \right)=x\) for \(0.001 < x < 1\)[2]