This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.2 Problem 3 Solution. Solutions for other problems are available at Exercise 1.2 Solutions
Exercise 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.
