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I have functions $f$ and $g$. However, the domain of $f(g)$ is not the same as the domain of $g$.

  • $f = \{ (1,2), (2,3), (3,4) \}$

  • $g = \{ (1,2), (2,0), (3,5), (4,3) \}$

  • $f(g) = \{ (1,3), (4,4) \}$

  • $g(f) = \{ (1,0), (2,5), (3,3) \}$

However, the domain of $f(g)$ does not equal the domain of $g$. So is $f(g)$ not defined in this case? Or is it a function with a smaller domain than $g$?

FishDrowned
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JacksonMadison
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    For short, $f(g)$ is still a function but with a smaller domain. Note that the function $f$ has domain ${1,2,3}$ and hence you have to make sure that whatever comes out of $g$ is in this set, to allow that $f(g)$ makes sense. Perhaps someone will elaborate on this for you, this is just some intuition. – xyz Nov 22 '23 at 19:08

1 Answers1

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The domain for a composite function $f(g)$ is not just the domain of $g$ but rather it is equal to: $$D_{f(g)}=\{x \in D_{g}\vert g(x) \in D_{f}\}$$ which means the domain of $f(g)$ is equal to the domain of $g$ if and only if $(\forall x \in D_{g} \implies g(x) \in D_{f})$ which isn't true for this case. We have $(1,4 \in D_g \implies g(1)=2,g(4)=3 \in D_{f})$

Hannan
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