The greatest integer function \( f:R\to R \) defined by \( f(x)=[x] \), then which of the following are true ?
(A) \( f(x) \) is one - one and onto.
(B) Domain of \( f(x)=R \) (the set of real numbers).
(C) \( f(x) \) is neither one-one nor onto.
(D) Range of \( f(x)=Z \) (The set of integers).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (C) and (D) only
\( [x] \) is the greatest integer less than or equal to x; it is defined for every real x and always takes an integer value, and each integer n equals [n].
\( \displaystyle f(1)=f(1.5)=1 \)
so f is not one-one. Also 0.5 is in the codomain R but is not the value of [x] for any x, so f is not onto.
So the domain of f is R, the range is Z, and f is neither one-one nor onto.
Checking the statements
(A) \( f(x) \) is one - one and onto. — Incorrect. f(1) = f(1.5) = 1, so f is not one-one.
(B) Domain of \( f(x)=R \) (the set of real numbers). — Correct. [x] is defined for every real number.
(C) \( f(x) \) is neither one-one nor onto. — Correct. f(1) = f(1.5), and 0.5 is never a value of [x].
(D) Range of \( f(x)=Z \) (The set of integers). — Correct. Every value is an integer, and each integer n equals [n].