Show that the function f : R∗ → R∗ defined by f(x) = 1/x is one-one and onto, where R∗ is the set of all non-zero real numbers. Is the result true, if the domain R∗ is replaced by N with co-domain being same as R∗ ?

Class 12, NCERT Chapter 1, Exercise 1.2, Q1

It is given that f: R∗ →R∗ is defined by f(x) =1/x
For one – one:
Let x, y ∈ R∗
such that f(x) = f(y)
⇒1/x =1/y
⇒ x = y
Therefore, f is one–one.
For onto:
For y ∈ R∗,

there exists x =1/y ∈ R∗ [as y ≠ 0]

such that
⇒f(x) = 1/ [1/y]= y
Therefore, f is onto.
Thus, the given function f is one-one and onto.
Now, consider function g:N→R∗ defined by g(x) =1/x
We have,
⇒g(x) = g(y)
⇒1/x=1/y
⇒x = y

Therefore, g is one–one.
Further, it is clear that g is not onto as for 1.2 ∈ R∗,

there does not exit any x in N
such that g(x) =1/1.2

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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