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A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
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
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
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