Because you got instruction to do (at least some of) it in Maple...
restart: g := s -> (1+2*s)/(1+s); # g is an operator (ie. function) 1 + 2 s s -> ------- 1 + s g(z); # g(z) is an expression 1 + 2 z ------- 1 + z isolate(g(z)=t, z); # isolate g(z) for z -1 + t z = - ------ -2 + t rhs(%); # an expression -1 + t - ------ -2 + t h1:=unapply(%, t); # h1 is also an operator -1 + t t -> - ------ -2 + t h1(g(z)); # composition of operators (functions) 1 + 2 z -1 + ------- 1 + z - ------------ 1 + 2 z -2 + ------- 1 + z simplify(%); # you've shown h1(g(z))=z z g(h1(z)); # composition of operators 2 (-1 + z) 1 - ---------- -2 + z -------------- -1 + z 1 - ------ -2 + z simplify(%); # h1 serves as desired h2, as well z