The Greatest Integer Function \(f : \mathrm{R} \to \mathrm{R}\) given by \(f(x) = [x], x \in \mathrm{R}\) and \([x]\) denotes the greatest integer less than or equal to x is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → neither one-one nor onto
Not one-one: different inputs give the same output. For example,
\( f(1) = [1] = 1 \)
\( f(1.5) = [1.5] = 1 \)
but \(1 \ne 1.5\).
Not onto: [x] is always an integer, so a real number such as 0.5 in the co-domain R is never an output. The range is the set of integers Z, not all of R.
So f is neither one-one nor onto.
NCERT (2026-27 edition):
Types of Functions
