Why is the domain of a composition of functions an intersection and not a union?

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I'm studying the composition of functions from the University of Toronto Pre-calculus website.

The instructions for determining the domain of a composition is as follows:

$(f \circ g)(x)$ is defined when $g(x)$ and $f(g(x))$ are defined! So when determining the domain of a composition of functions, take the domain of the inner function, and continue outwards to see if you need to remove any more $x$-values. I.e., $domain((f\circ g)(x)) = domain(g(x)) \cap domain(f(g(x)))$.

My question: why is the domain of a composition of function the intersection of the constituent functions? Shouldn't it be the union? Doesn't taking the intersection mean that if one function has a restriction not shared by the other, it won't apply to the composition? And wouldn't that be a bad thing?

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2 Answers

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It means the intersection of the domains (the non-restricted inputs), rather than the intersection of the restrictions!

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Ah, nevermind. I think I see my error now.

I was thinking about the domain as the set of values that are to be excluded, rather than the set of values that are to be allowed.

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