Question 1037620
Both functions are linear in x.


Expect g(x) to be a linear function in x, and g might be kx+p, so start it as  {{{g(x)=kx+p}}}.


{{{g(f(x))=k(f(x))+p}}}
{{{g(f(x))=k(1+2x)+p}}}
{{{g(f(x))=k+2kx+p}}}
{{{g(f(x))=2kx+k+p}}}


Given is the composition {{{g(f(x))=x+3}}}.


The two composition functions must be equal.
{{{2kx+k+p=x+3}}}



Identify and equate the corresponding parts.
{{{system(2k=1,k+p=3)}}}
-
{{{system(k=1/2,k+p=3)}}}


{{{p=3-k}}}
{{{p=3-1/2}}}
{{{p=2&1/2}}}



Return to the more variablized function {{{g(x)=kx+p}}}, and substitute the values found.
{{{highlight(g(x)=x/2+2&1/2)}}}