Crazy function

$we\quad observe\quad that\quad f(2x)-f(x)=1\quad for\quad 1,2,4,8,16.\\ donote\quad g(x)=f(2x)-(x)-1\quad for\quad 1,2,4,8,16\\ g(x)=0\quad when\quad x=1,2,4,8,16\\ \Rightarrow g(x)=a(x-1)(x-2)(x-4)(x-8)(x-16),\quad where\quad a\quad is\quad constant\\ when\quad x\quad =0\\ g(0)=a(-1)(-2)(-4)(-8)(-16)\\ -1=a(-1)(-2)(-4)(-8)(-16)\\ a=\frac { 1 }{ 1\times 2\times 4\times 8\times 16 } \\ g(x)=\frac { (x-1)(x-2)(x-4)(x-8)(x-16) }{ 1\times 2\times 4\times 8\times 16 } \\ The\quad coefficient\quad of\quad x\quad in\quad g(x)\quad is\quad \frac { 1 }{ 1 } +\frac { 1 }{ 2 } +\frac { 1 }{ 4 } +\frac { 1 }{ 8 } +\frac { 1 }{ 16 } =\frac { 31 }{ 16 } (\ast )\\ (\ast )for\quad people\quad who\quad do\quad not\quad understand\quad only.\\ (x\quad term\quad in\quad g(x)\quad is\quad formed\quad by\quad timing\quad four\quad number\quad from\quad 1,2,4,8,16\quad and\quad x\\ \quad \Rightarrow \quad \frac { 1\times 2\times 4\times 8 }{ 1\times 2\times 4\times 8\times 16 } x\quad +\frac { 1\times 2\times 4\times 16 }{ 1\times 2\times 4\times 8\times 16 } x\quad +\frac { 1\times 2\times 8\times 16 }{ 1\times 2\times 4\times 8\times 16 } x+\frac { 1\times 4\times 8\times 16 }{ 1\times 2\times 4\times 8\times 16 } x+\frac { 2\times 4\times 8\times 16 }{ 1\times 2\times 4\times 8\times 16 } x\\ \Rightarrow The\quad coefficient\quad of\quad x\quad in\quad g(x)\quad is\quad \frac { 1 }{ 1 } x+\frac { 1 }{ 2 } x+\frac { 1 }{ 4 } x+\frac { 1 }{ 8 } x+\frac { 1 }{ 16 } x=\frac { 31 }{ 16 } x)\\ \\ for\quad the\quad coefficient\quad of\quad x\quad term\quad in\quad g(x)\quad is\quad same\quad as\quad f(x).\quad \\ Therefore\quad answer\quad is\quad 31+16=47$