Question

If g and f(g(x)) are both onto, does it follow that f is onto? If g and f(g(x)) are both onto, does it follow that f is onto? @Mathematics

Answer 1

Yes. In fact it is sufficient that (f o g)(x) = f(g(x)) is onto.

Proof:
I'll write these are functions U --> U, for some set U.

As (f o g) is also onto, for all y in U, there is an x such that
f(g(x)) = y. Now let w = g(x).

Now as y was arbitrary, it follows that for every such y there is a number w such that
f(w) = y.

Answer 2

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Example
g(x) = sin x
f(x) = tan(x.pi/2)

g is only onto [-1,1], not the entire reals.

But f(g(x)) = tan(sin x . pi/2) is onto the entire real line.