Why does f[g(x)] exist only when the range of g(x) overlaps the domain of f(x)?

Question

For context, this is in a year 12 advanced functions course talking about combining functions.

Answer 1

A function $f(x)$ maps points $x$ to other values. But this may only be defined for certain values. The domain is what $f$ maps FROM, and the range is what $f$ maps TO.

You have $f(g(x))$, so the $f$ will be mapping values of $g(x)$, so the range of $g(x)$ must be in the domain of $f$ for this to make sense.

EDIT: Think if $f$ maps $1 \to 2, $ and $3 \to 4$.

Then $f(g(x))$ Can only have a solution if $g(x)$ takes $x$ to $1$ or $3$.