Show that the function f : N→ N, given by f (1) = f(2) = 1 and f(x) = x – 1, for every x

Show that the function f : N→ N, given by f (1) = f(2) = 1 and f(x) = x – 1, for every x > 2, is onto but not one-one.

Class 12, NCERT Chapter 1, Example9

f is not one-one,

as f(1) = f(2) = 1. [ it has two preimages ]

But f is onto,

as given any y ∈ N, y ≠ 1,

we can choose x as y + 1

such that f(y + 1) = y + 1 – 1 = y.

Also for 1 ∈ N, we have f(1) = 1.

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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