GATE | Gate IT 2005 | Question 30

Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a ∈ A. Which of the following statements is always true for all such functions f and g?

(A) g is onto => h is onto
(B) h is onto => f is onto
(C) h is onto => g is onto
(D) h is onto => f and g are onto

Answer: (C)

Explanation: g(f(a)) is a composition function, which is $A \rightarrow B \rightarrow C$ . If h: $A \rightarrow C$ is onto, the composition must be onto, but the first function in the composition need not to be onto (surjective function), so g: $B \rightarrow C$ must be onto.
So, option (C) is correct.