If f(x) and g(x) are onto (surjective) functions, which ONE of the following is TRUE about the function g(f(x))? A. g(f(x)) must be an onto (surjective) function. B. g(f(x)) must be a one-to-one (injective) function. C. g(f(x)) must be a bijective function, that is, both one-to-one and onto. D. g(f(x)) is not required to be a one-to-one or onto function.

GATE 2025 · Discrete Mathematics · Onto · medium

Answer: D. g(f(x)) is not required to be a one-to-one or onto function.

1. Verify surjectivity of g o f: Since f is onto B, every b = f(a) for some a. Since g is onto C, every c = g(b) for some b. Combining: c = g(f(a)), so g o f is onto C.
2. Disprove injectivity with a counterexample: Let A={1,2,3}, B={a,b}, C={x,y}. f(1)=f(2)=a, f(3)=b (onto B). g(a)=x, g(b)=y (bijective). Then g(f(1))=g(f(2))=x, so g o f is NOT injective. Options B and C are false. D (not required to be one-to-one) is the correct answer.