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
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
$\endgroup$ 21 Answer
$\begingroup$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 $g(x)\neq0$.
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