Let \(f: \mathrm{R} \rightarrow \mathrm{R}\) be defined as \(f(x) = [x]\), where \([x]\) denotes the greatest integer less than or equal to \(x\). Then which of the following statements are correct?
(A) \(f\) is one-one but not onto
(B) \(f\) is not onto
(C) \(f\) is not one-one
(D) \(f\) is one-one and onto
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B) and (C) only
The greatest integer function takes the same value on a whole interval:
\(f(1.2) = f(1.8) = 1\), although \(1.2 \ne 1.8\)
Its values are integers only, so its range is the set of integers, not all of R:
\([x] = 0.5\) has no solution
So f is not one-one and not onto.
Checking the statements
(A) \(f\) is one-one but not onto — Incorrect. f is not one-one: \(f(1.2) = f(1.8)\).
(B) \(f\) is not onto — Correct. 0.5 is not a value of f.
(C) \(f\) is not one-one — Correct. \(f(1.2) = f(1.8) = 1\).
(D) \(f\) is one-one and onto — Incorrect. f is neither.
NCERT (2026-27 edition):
Types of Functions
