Question 733563
"find the value of g(f(x+1))", not certain this is properly asked.  You need specific information about some numbers to find a value.  Here is what we can do with those two functions about the composition.


From g(x),  g(x)=x+3.  We want to let x=f(x).
g(__)=(__)+3, put f(x) in the grouping location.  {{{g(f(x))=f(x)+3}}}.


We already have a definition for f(x), so we will use this expression:
{{{g(f(x))=(x^2-2)+3}}}.  BUT THAT IS STILL NOT FINISHED.  We are asked for not simply {{{x=f(x)}}}, but we are asked for {{{x=f(x+1)}}}.  Then we want:


{{{g(f(x+1))=f(x+1)+3}}}
{{{g(f(x+1))=((x+1)^2-2)+3}}}
={{{(x^2+2x+1 -2)+3}}}
={{{x^2+2x-1+3}}}
={{{x^2+2x+4}}}


{{{highlight(g(f(x+1))=x^2+2x+4)}}}