Domain and Range of a Composition

Think of two machines wired in series. The first machine, g, takes in x and produces an output. That output feeds straight into the second machine, f. For the whole pipeline to work, x has to be legal for g, but that is not enough — the number g spits out has to be legal for f too. If f chokes on some value that g happily produces, that x gets kicked out of the domain of the composition, even though g itself was fine with it.

This is why composing functions can shrink the set of usable inputs. g might accept all real numbers, but if f is something like 1/x or √x, f is picky about what it accepts. Any x that makes g(x) land on a number f rejects — like 0 for a reciprocal, or a negative number for a square root — has to be excluded, even though g never complained. So the rule is: screen x against g's domain first, then take g(x) and screen that against f's domain. Both screens have to pass.

This connects directly to how composition itself is built: you always evaluate the inner function first, then feed the result into the outer one, so the domain logic follows the same inside-out order. It also sets up inverse functions, where swapping input and output roles means swapping domain and range, and transformations, where shifting or stretching a parent graph can shift or stretch its domain and range the same way.