We can view the domain and range of a function as separate inputs and outputs of a box processor:
Domain of f = D f
Range of f = R f
AKA
Input to box
f ( x)
AKA
Box
Output of box
We can also see a composite function in a similar fashion. For example, considering a generic
composite function gf ( x), the following system can be imagined:
Domain of f = D f
Range of f = R f now
is also domain of gf
becomes domain of g
AKA
f ( x)
Input to box 1 is
also input to overall
g ( x)
AKA
Output of box 1 is fed into
Box 1
box 2 as its input
Box 2
system ( boxes 1 & 2)
Range of gf
System box of composite function gf (blue dotted)
Output of
AKA
system
1. The domain of f ( x ) is plugged into the system gf ( x ) as the interfacing input, hence D f = D gf .
2. Within the system gf (x ) (blue dotted), the output of f (x ) = R f is subsequently fed as an
input into g (x ) ; ie D g = R f .
3. The output of g (x ) is also the output of the entire system gf (x ) , ie Rgf .