Let f: mathbbR → mathbbR be defined as f(x) = [x], where [x] denotes the greatest integer less than or equal to x. Then which of the following…

The function $f(x) = [x]$f(x)=[x] is the greatest integer function (or floor function), where $[x]$[x] gives the largest integer that is less than or equal to $x$x.

Examples:
$$[2.7] = 2$$[2.7]=2
$$[5] = 5$$[5]=5
$$[-1.3] = -2$$[−1.3]=−2

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Checking if $f$f is one-one:

A function is one-one if different inputs always give different outputs.

Take $x_1 = 1.2$x1​=1.2:
$$f(1.2) = [1.2] = 1$$f(1.2)=[1.2]=1

Take $x_2 = 1.7$x2​=1.7:
$$f(1.7) = [1.7] = 1$$f(1.7)=[1.7]=1

Here $x_1 \neq x_2$x1​=x2​ but $f(x_1) = f(x_2)$f(x1​)=f(x2​).

Therefore, $f$f is not one-one.

Statement (C) is correct.

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Checking if $f$f is onto:

A function $f: \mathbb{R} \to \mathbb{R}$f:R→R is onto if every real number is an output.

The output of $[x]$[x] is always an integer: $..., -2, -1, 0, 1, 2, ...$...,−2,−1,0,1,2,...

For example, there is no $x$x where $[x] = 1.5$[x]=1.5.

Therefore, $f$f is not onto.

Statement (B) is correct.

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Since $f$f is not one-one, statement (A) is incorrect.

Since $f$f is neither one-one nor onto, statement (D) is incorrect.

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The correct statements are (B) and (C).