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**NCERT mathematics class 12 chapter Relations and Functions Exercise 1.2 Problem 3 Solution**. Solutions for other problems are available at Exercise 1.2 SolutionsExercise 1.2 Problem 3 Solution

3. Prove that the Greatest Integer Function f : R → R, given by {f(x) = [x]}, is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

To check whether f is one-one:

The function f : R → R is defined as {f(x) = [x]}

For instance,

f(2) = [2] = 2

f(2.34) = [2.34] = 2

f(2.999) = [2.999] = 2

f(2.578344) = [2.578344] = 2

So, the same element 2 is image of many different elements in the domain.

⇒ Every integer element in the co-domain R is an image of multiple elements in the domain R.

∴ f is not one-one.

To check whether f is onto:

The function f : R → R is defined as {f(x) = [x]}

This means that every element in the domain is mapped to an integer in the co-domain.

⇒ The elements in the co-domain which are not integers are not images of any element in the domain.

For instance, 1.23 is not an image of any element in the domain.

∴ f is not onto.