g(.) is a function from A to B, f(.) is a function from B to C, and their composition f(g(.)) is a m

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Question

g(.) is a function from A to B, f(.) is a function from B to C, and their composition 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(.)?

Accepted Answer

Correct Answer: (D) g(.) is not required to be one-to-one or onto.Concept — Surjectivity of Function Composition:A function h: X → Y is surjective (onto) if every element of Y is the image of at least one element of X. In other words, the range of h equals its codomain Y.Given: f: B → C is surjective, and f∘g: A → C is surjective.Key question: What properties must g: A → B have?Analysis: For f∘g to be surjective onto C, for every c ∈ C, there must exist some a ∈ A such that f(g(a)) = c. Since f is surjective, for every c ∈ C there exists some b ∈ B such that f(b) = c. Now, we only need g to "reach" those specific elements b — but g does not need to reach every element of B. g may miss some elements of B that are redundant for f's output.Counterexample showing g need not be surjective or injective:Let A = {1, 2}, B = {3, 4, 5}, C = {6}.Define g: g(1) = 3, g(2) = 4. (g is not surjective — 5 ∈ B is not reached; g is injective here.)Define f: f(3) = 6, f(4) = 6, f(5) = 6. (f is surjective.)Then f∘g: f(g(1)) = f(3) = 6, f(g(2)) = f(4) = 6. f∘g is surjective onto C = {6}. ✓Now change g to: g(1) = 3, g(2) = 3. (g is not injective either.) f∘g is still surjective. ✓This confirms g need not be surjective or injective. Option D is correct.Note on Option A: If f∘g is surjective but g is NOT surjective, f must still be surjective over the full B (which it is given). So g's surjectivity onto B is not required — only that f covers C through whatever image g produces.

g(.) is a function from A to B, f(.) is a function from B to C, and their composition 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(.)?

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g(.) is a function from A to B, f(.) is a function from B to C, and their composition 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(.)?

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