If f: R→ R is a function given by f(x) = [x] (greatest integer function), then which of the following is/are correct

(a) fis one-one

(b) f is not onto

(c) Range of f is I (set ofthe integers)

(d) f (2.5) = 2

(e). f is bijective

Choose the correct answer from the options given below:

The correct answer is B, C, D only.

Let us analyze the given function $f:ℝ\to ℝ$ defined by $f\left(x\right)=\left[x\right]$, where $\left[x\right]$ represents the greatest integer function (which outputs the greatest integer less than or equal to $x$).

Let's evaluate each statement step-by-step:

Statement (a): $f$ is one-one
A function is one-one (injective) if distinct inputs yield distinct outputs. Let's test this with two different real numbers, say $x_1=2.1$ and $x_2=2.5$:
$f(2.1)=[2.1]=2$
$f(2.5)=[2.5]=2$
Since $2.1\ne 2.5$ but $f(2.1)=f(2.5)$, the function is not one-one. Therefore, statement (a) is incorrect.

Statement (b): $f$ is not onto
A function $f:ℝ\to ℝ$ is onto (surjective) if its range is equal to its codomain ($ℝ$). The outputs of the greatest integer function are always integers (set $\mathbb{Z}$ or $I$). Non-integer real numbers in the codomain, such as $1.5$, have no pre-image in the domain because $\left[x\right]$ can never equal $1.5$. Since the range ($\mathbb{Z}$) is a proper subset of the codomain ($ℝ$), the function is not onto. Therefore, statement (b) is correct.

Statement (c): Range of $f$ is $I$ (set of the integers)
By definition, the greatest integer function $\left[x\right]$ maps every real number to the nearest integer less than or equal to it. The set of all possible outputs is indeed the set of all integers, denoted by $I$ (or $\mathbb{Z}$). Therefore, statement (c) is correct.

Statement (d): $f(2.5)=2$
Evaluating the function at $x=2.5$:
$f(2.5)=[2.5]$
The greatest integer less than or equal to $2.5$ is $2$. Thus, $f(2.5)=2$. Therefore, statement (d) is correct.

Statement (e): $f$ is bijective
A function is bijective if it is both one-one and onto. Since we have shown that $f$ is neither one-one nor onto, it is not bijective. Therefore, statement (e) is incorrect.

Combining our findings, only statements B, C, and D are correct.