A function f from the set of natural numbers to integers defined by f(n)=n−12, when n is odd−n2, when n is even is

Let x1 and x2 be any two elements in the domain (N).Case 1: When both x1 and x2 are even.Let f(x1)=f(x2)⇒−x12=x22⇒−x1=x2⇒x1=x2Case 2 : When both x1 and x2 are odd.Let f(x1)=f(x2)⇒x1−12=x2−12⇒x1=x2Case 3 : When x1 be even and x2 be odd.Then, f(x1)=−x12 and f(x2) = x2−12Then clearly, ⇒x1≠x2⇒f(x1)≠f(x2)From all the cases, we can say that , f is one−one.⇒Co−domain of f=Range of f∴f is onto.

A function f from the set of natural numbers to integers defined by f(n)=n−12, when n is odd−n2, when n is even is

Let x1 and x2 be any two elements in the domain (N).Case 1: When both x1 and x2 are even.Let f(x1)=f(x2)⇒−x12=x22⇒−x1=x2⇒x1=x2Case 2 : When both x1 and x2 are odd.Let f(x1)=f(x2)⇒x1−12=x2−12⇒x1=x2Case 3 : When x1 be even and x2 be odd.Then, f(x1)=−x12 and f(x2) = x2−12Then clearly, ⇒x1≠x2⇒f(x1)≠f(x2)From all the cases, we can say that , f is one−one.⇒Co−domain of f=Range of f∴f is onto.

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A function f from the set of natural numbers to integers defined by $f (n) = \{\begin{cases} \frac{n - 1}{2}, w h e n n i s o d d \\ \frac{- n}{2}, w h e n n i s e v e n \end{cases} i s$

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Neither one – one nor onto

one-one but not onto

Onto but not one-one

one-one and onto both

answer is D.

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Detailed Solution

$\begin{array}{l} L e t x_{1} a n d x_{2} b e a n y t w o e l e m e n t s i n t h e d o m a i n (N) . \\ C a s e 1 : W h e n b o t h x_{1} a n d x_{2} a r e e v e n . \\ L e t f (x_{1}) = f (x_{2}) \\ \Rightarrow \frac{- x_{1}}{2} = \frac{x_{2}}{2} \\ \Rightarrow - x_{1} = x_{2} \\ \Rightarrow x_{1} = x_{2} \\ C a s e 2 : W h e n b o t h x_{1} a n d x_{2} a r e o d d . \\ L e t f (x_{1}) = f (x_{2}) \\ \Rightarrow \frac{x_{1} - 1}{2} = \frac{x_{2} - 1}{2} \\ \Rightarrow x_{1} = x_{2} \\ C a s e 3 : W h e n x_{1} b e e v e n a n d x_{2} b e o d d . \\ T h e n, f (x_{1}) = \frac{- x_{1}}{2} a n d f (x_{2}) = \frac{x_{2} - 1}{2} \\ T h e n c l e a r l y, \\ \Rightarrow x_{1} \neq x_{2} \\ \Rightarrow f (x_{1}) \neq f (x_{2}) \\ F r o m a l l t h e c a s e s, w e c a n s a y t h a t, f i s o n e - o n e . \\ \Rightarrow C o - d o m a i n o f f = R a n g e o f f \\ ∴ f i s o n t o . \end{array}$