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(·)?

g(·) is not required to be a one-to-one or onto function.

g(·) must be an onto (surjective) function.

g(·) must be a one-to-one (injective) function.

g(·) must be a bijective function, that is, both one-to-one and onto.

g(·) is not required to be a one-to-one or onto function.

GATE CSE 2025 Session 1 Q17: What f and f∘g onto imply about g

Engineering Mathematics
1 mark
MCQ
−1/3 if wrong

$g(\cdot)$ is a function from $A$ to $B$ , $f(\cdot)$ is a function from $B$ to $C$ , and their composition defined as $f(g(\cdot))$ is a mapping from $A$ to $C$ .

If $f(\cdot)$ and $f(g(\cdot))$ are onto (surjective) functions, which ONE of the following is TRUE about the function $g(\cdot)$ ?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: D (official GATE 2025 final answer key)

Concept used: If $f \circ g$ is onto, only $f$ is forced to be onto; $g$ can be anything.

First step: Try a counterexample: $A = \{1, 2\}$ , $B = \{x, y, z\}$ , $C = \{c\}$ , $g(1) = g(2) = x$ ; then $f$ and $f \circ g$ are onto but $g$ is neither one-to-one nor onto.

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Source: official GATE 2025 Session 1 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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GATE CSE 2025 Session 1 Q17: What f and f∘g onto imply about g

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