If R denotes the set of all real numbers then the function f : R → R defined f (x) = [x] isOne-one onlyOnto onlyBoth one-one and ontoNeither one-one nor onto

The function f(x) = [x] is defined as the greatest integer function or the floor function. It gives the greatest integer less than or equal to x.

1. One-One (Injective) Function: A function is said to be one-one (or injective), if the images of different elements under the function are different, i.e., for every x and y in the domain, if x ≠ y, then f(x) ≠ f(y).

In the case of f(x) = [x], if we take two different real numbers, say 1.1 and 1.2, both of them give the same image under the function, which is 1. Therefore, the function f(x) = [x] is not one-one.

2. Onto (Surjective) Function: A function is said to be onto (or surjective), if every element in the co-domain has a pre-image in the domain.

In the case of f(x) = [x], every real number in the co-domain has a pre-image in the domain. For example, the number 2 in the co-domain has a pre-image in the domain, which could be any number in the interval [2, 3). Therefore, the function f(x) = [x] is onto.

So, the function f(x) = [x] is not one-one but it is onto.