dominated range, continuous version

Let R denote the reals, and let proj_x be the projection map from R^2
to R onto the x-coordinate (proj_x(x,y)=x).

Let f,g be continuous functions from R to R such that range(f) is a
subset of range(g).

Thus, the range of g "dominates" the range of f.

Conjecture 1:

There exists a continuous function h from R to R such that f = g o h.

Assuming conjecture 1 is false, then consider the following weaker
claim ...

Conjecture 2:

Let S={(x,y) in R^2 | f(x)=g(y)}. Then, for some component C of S,
proj_x(C)=R.

Remarks:

I don't really believe either of the above conjectures.

If you prefer to just give a hint for a proof or disproof, that's
fine. Of course, a solution would also be fine.

quasi
.

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