$f : N \to N, w h e r e f (x) = x - {(- 1)}^{x} t h e n f i s$

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One one and onto

Many one and into

Many one and onto

One one and into

answer is C.

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Detailed Solution

$\begin{array}{l} f : N \to N a n d f (x) = x - {(- 1)}^{x} \\ f (x) = {\begin{cases} x - 1 i f x i s e v e n \\ x + 1 i f x i s o d d \end{cases} \\ f^{1} (x) = {\begin{cases} 1 i f x i s e v e n \\ 1 i f x i s o d d \end{cases} \\ ∴ f^{1} (x) > 0, i n c r e a \sin g f u n c t i o n \\ ∴ f i s o n e - o n e \\ f (N) = {\begin{cases} 1, 3, 5 - - - i f x i s e v e n \\ 2, 4, 6 - - - i f x i s o d d \end{cases} \\ ∴ R a n g e = {1, 2, 3 - - - n} = c o d o m a i n \\ f i s o n t o \end{array}$