Bijective Function

Correct Answer: (A)

Why (A) is true: Since f is a bijection onto [0, ∞), every value in [0, ∞) — including 0 — must be achieved. If f is monotone increasing, it is order-preserving, so f(0) ≤ f(x) for all x ≥ 0. For f to be surjective onto [0, ∞), its minimum value must be 0, forcing f(0) = 0.

Why (R) is true and explains (A): The Reason correctly identifies both pillars of the proof — (i) an increasing bijection on [0, ∞) maps the boundary point 0 to 0, and (ii) a decreasing bijection is impossible because [0, ∞) is bounded below but unbounded above, so a decreasing map onto [0, ∞) cannot be surjective. Together these two facts constitute the complete and correct explanation of the Assertion.