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.

Class 12, NCERT Chapter 1, Exercise 1.2, Q3

f:R→R given by,

f(x) = [x]

It is seen that f(1.2) = [1.2] = 1, f(1.9) = [1.9] = 1. ,

Therefore f(1.2) = f(1.9), but 1.2 is not equal to 1.9. ,

therefore f is not one-one.

Now, consider 0.7 ∈ R.

It is known that f(x)= [x] is always an integer. Thus, there does not exist any element x ∈ R such that f(x) = 0.7. ,

therefore f is not onto.

Hence, the greatest integer function is neither one-one nor onto.

Comments

. Let f : R → R be defined as f(x) = 3x. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.

Class 12, NCERT Chapter 1, Exercise 1.2, Q12 f : R → R defined as f ( x ) = 3 x . Let x , y ∈ R such that f ( x ) = f ( y ). ⇒ 3 x = 3 y ⇒ x = y ∴ f is one-one. Also, for any real number ( y) in co-domain R , there exists in R such that . ∴ f is onto. Hence, function f is one-one and onto. The correct answer is A.

Let f : N → N be defined by f (n) ={ (n+1)/2, if n is odd and (n-1)/2, if n is even, for all n ∈ N.State whether the function f is bijective. Justify your answer

Class 12, NCERT Chapter 1, Exercise 1.2, Q9 Thus it is bijective.

Show that a one-one function f : {1, 2, 3} → {1, 2, 3} must be onto.

Class 12, NCERT Chapter 1, Example14 Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different elements of the co-domain {1, 2, 3} under f. Hence, f has to be onto .

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