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g(.) is a function from A to B, f(.) is a function from B to C, and their composition defined as f(g(.)) is a mapping from A to C. If f(.) and f(g(.)) are onto (surjective) functions, which ONE of the following is TRUE about the function g(.)?

1. g(.) must be an onto (surjective) function.
2. g(.) must be a one-to-one (injective) function.
3. g(.) must be a bijective function, that is, both one-to-one and onto.
4. g(.) is not required to be a one-to-one or onto function.

Correct answer: g(.) is not required to be a one-to-one or onto function.

Solution

The correctness of the option lies in the fact that the surjectivity of the composition f(g(.)) does not impose any restrictions on g(.) being onto or one-to-one. As long as f(.) maps the outputs of g(.) to all elements in C, g(.) can have any mapping characteristics.

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