SOLUTION: If f(x) and f(g(x)) are surjective does it follow that g(x) is surjective?

Question 504558: If f(x) and f(g(x)) are surjective does it follow that g(x) is surjective?
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Recall that if h(x) is surjective, then every element in the range of h(x) is mapped to by at least one x ∈ S. Here we have to assume that the domain and range are all real numbers.

Suppose that g(x) is not surjective, that is, there is a real number in g that is not mapped onto (e.g. if g(x) = |x|, -3 would be a counterexample). Then f(g(x)) would not be defined everywhere because g(x) has an incomplete domain so f(g(x)) would not be surjective over all real numbers (another way to phrase it -- the cardinality of the range of f(g(x)) is less than or equal to the cardinality of the domain of g(x), so there must be some f(g(x)) value that is not mapped onto). This contradicts our claim so g(x) must be surjective.
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