The function f(x)=[x] is the greatest integer function (or floor function), where [x] gives the largest integer that is less than or equal to x.
Examples:
[2.7]=2
[5]=5
[−1.3]=−2
Checking if f is one-one:
A function is one-one if different inputs always give different outputs.
Take x1=1.2:
f(1.2)=[1.2]=1
Take x2=1.7:
f(1.7)=[1.7]=1
Here x1=x2 but f(x1)=f(x2).
Therefore, f is not one-one.
Statement (C) is correct.
Checking if f is onto:
A function f:R→R is onto if every real number is an output.
The output of [x] is always an integer: ...,−2,−1,0,1,2,...
For example, there is no x where [x]=1.5.
Therefore, f is not onto.
Statement (B) is correct.
Since f is not one-one, statement (A) is incorrect.
Since f is neither one-one nor onto, statement (D) is incorrect.
The correct statements are (B) and (C).