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A function $$f$$ from the set of natural numbers to integers defined by $$$f\left( n \right) = \left\{ {\begin{matrix} {{{n - 1} \over 2},\,when\,n\,is\,odd} \\ { - {n \over 2},\,when\,n\,is\,even} \\ \end{matrix} } \right.$$$ is

JEE · Math · previous-year question

  1. A.neither one -one nor onto
  2. B.one-one but not onto
  3. C.onto but not one-one
  4. D.one-one and onto bothcorrect

Answer

D. one-one and onto both

Explanation

We have $$f:N \to I$$ If $$x$$ and $$y$$ are two even natural numbers, then $$f\left( x \right) = f\left( y \right) \Rightarrow {{ - x} \over 2} = {{ - y} \over 2} \Rightarrow x = y$$ Again if $$x$$ and $$y$$ are two odd natural numbers then $$f\left( x \right) = f\left( y \right) \Rightarrow {{x - 1} \over 2} = {{y - 1} \over 2} \Rightarrow x = y$$ $$\therefore$$ $$f$$ is onto. Also each negative integer is an image of even natural number and each positive integer is an image of odd natural number. $$\therefore$$ $$f$$ is onto. Hence $$f$$ is one one and onto both.

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