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

Solution:

Greatest Integer Function is a function that gives the greatest integer less than or equal to the number. The greatest integer less than or equal to a number x is represented as ⌊x⌋.

According to the given problem.

f : R → R given by f (x) = [x]

f (1.2) = [1.2] = 1,

f (1.9) = [1.9] = 1

⇒ f (1.2) = f (1.9),

but 1.2 ≠ 1.9

⇒ f is not one-one.

Consider 0.7 ∈ R

f (x) = [x] is an integer.

There does not exist any element x ∈ R such that f (x) = 0.7

⇒ f is not onto.

The greatest integer function is neither one-one nor onto

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.2 Question 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.

Summary:

f is not one-one. There does not exist any element x ∈ R such that f (x) = 0.7. f is not onto. The greatest integer function is neither one-one nor onto