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