If [tex]f(g(x))=g(f(x))=x[/tex], then [tex]g[/tex] and [tex]f[/tex] are inverses of one another. So we know [tex]g(f(4))=4[/tex].
[tex]g(f(4))=2+\ln(f(4)+1)=4[/tex]
[tex]\implies\ln(f(4)+1)=2[/tex]
[tex]\implies f(4)+1=e^2[/tex]
[tex]\implies\boxed{f(4)=e^2-1}[/tex]
