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Let $f$ and $g$ be functions. Then the domain of $f$ composed with $g$ is the intersection of the domain of $f$ and the domain of $g$. True or False?

I feel that the answer should be false. Consider $g$ maps the positive reals to the negative reals and $f$ maps the negative reals back to the positive reals. It certainly seems to me that the domain of $f$ composed with $g$ (equivalently, $f(g(x))$?) would simply be the positive reals rather than the intersection of the negative and positive reals (which would give the empty set, I believe). However the answer key indicates that this statement is true. Can someone please explain the error in my reasoning?

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1 Answer

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The result as stated is clearly false. The domains of $f$ and $g$ need not even be subsets of the same universal set.

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