domain of composite function (is there a set rule)

Question

In the function topic of "function combinations" or "function algebra", for the basic arithmetic operations of the following:

1) f + g

2) f - g

3) f * g

4) f / g

To find the domain of these, one needs to find the domains of each f, g and find the intersection to get the domain.

I recently came across a problem asking about domain of a composite function. My book does not address this by giving any rules, and when I try a couple of problems the result of those problems is that it seems to be the INTERSECTION of the domains of the functions f and g.

I looked at the posts here, and could not find the answer to this, closest references to my question that I could find are these 2 links:

Domain and Range in function composition

Domain of composite functions

Hope someone can tell me if the domain of composite function is just the intersection of the 2 individual function domains. IF that is not the case, then can someone provide a counter-example.

Regards,

P

Answer 1

Bear in mind that $$(f\circ g)(x):=f\bigl(g(x)\bigr).$$ In order for $g(x)$ to mean anything, we must have $x\in\operatorname{dom}(g),$ and if $g(x)$ is meaningful, then in order for $f\bigl(g(x)\bigr)$ to mean anything, we must have $g(x)\in\operatorname{dom}(f).$ Consequently, we can say that $$\operatorname{dom}(f\circ g)=\bigl\{x\in\operatorname{dom}(g):g(x)\in\operatorname{dom}(f)\bigr\}.$$

In particular, then, the domain of $f\circ g$ is a subset of the domain of $g,$ consisting of those $x$ for which $g(x)$ is in the domain of $f$. (In a set-theoretic context, we often require that the domain of $f$ should be a codomain of $g$ before we talk about $f\circ g$ at all, in which case the domain of $f\circ g$ is just the domain of $g.$ This is not a universal convention, though, so be careful in using it reflexively.)

Incidentally, we have $$\operatorname{dom}(f/g)=\bigl\{x\in\operatorname{dom}(f)\cap\operatorname{dom}(g):g(x)\neq0\bigr\},$$ since even if $f(x)$ and $g(x)$ are defined, we can only say that $f(x)/g(x)$ is defined if $g(x)\neq0$.