Let $f: R→R$ 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
Given: $f:\mathbb{R} \to \mathbb{R}$ defined by $f(x) = [x]$, the greatest integer function.
Check one-one (injective): For $x_1 = 1.2$, $x_2 = 1.8$, $f(x_1) = f(x_2) = 1$, but $x_1 \ne x_2$. So $f$ is not one-one → (C) Correct
Check onto (surjective): For $y = 0.5 \in \mathbb{R}$, there is no $x \in \mathbb{R}$ such that $[x] = 0.5$. So $f$ is not onto → (B) Correct
Statements (A) and (D) are incorrect.