how to singularize existing function?

any group theory can describe the singularity function?

assume f and g are unknown

and assume solve(f, x) = solve(g, x)

f -> a

g -> a

b -> f

b ->g

if assume f = (x+1)*(x+2), g = (x+2)*(x+3)

and a = (x+1)*(x+2)*(x+3)

would like to find map from (x+1)*(x+2) to (x+1)*(x+2)*(x+3)

is it the solution subs(x=(x+1)*(x+2),(x+1)*(x+2)*(x+3)) by composition?

subs(x=(x+1)*(x+2),(x+1)*(x+2)*(x+3))

subs(x=1, (x+1)*(x+2));
subs(x=2, (x+1)*(x+2));
subs(x=1, (x+1)*(x+2)*(x+3));
subs(x=2, (x+1)*(x+2)*(x+3));

6 -> 24
12 -> 60

subs(x=1, ((x+1)*(x+2)+1)*((x+1)*(x+2)+2)*((x+1)*(x+2)+3)); # not 24
subs(x=2, ((x+1)*(x+2)+1)*((x+1)*(x+2)+2)*((x+1)*(x+2)+3)); # not 60

it seems composition is wrong

more difficult and general case should be

f(x,t)  -> a(x,t)

g(x,t)  -> a(x,t)

b(x,t) -> f(x,t)

b(x,t) -> g(x,t)

solve(f(x,t), x) = solve(g(x,t), x) = in terms of t

ex := taylor(exp(x), x = 0, 5);

it can not use with subs(x=1,ex);

how to remove O(x^5) part

want to write in maple code

to generate all commutative diagram 

with adjacency matrix

however, i only know a -> b, b-> c , a->d , d-> c

google no information about all commutative diagram, 

another problem is

would like to enrich theory , however, do not know how to connect property such as equations with diagram

(1/2)*(-x-y+sqrt(-3*x^2-6*x*y-3*y^2))/(x+y)

the complex value is always a constant whatever x and y 

i suspected that this is a wrong function