The function f : N → I defined by f x = x - 5 x 5 , where N is the set of natural numbers and x denotes the greatest integer less than or equal to x , is:

Question

The function f : N → I defined by f x = x - 5 x 5 , where N is the set of natural numbers and x denotes the greatest integer less than or equal to x , is:, one-one but not onto, one-one and onto, neither one-one nor onto, onto but not one-one

MARKS App · Accepted Answer

Given, f ( x ) = x − 5 [ x 5 ] Hence, f ( x ) = 5 ( x 5 − [ x 5 ] ) = 5 { x 5 } Since, x is a natural number. Hence, x 5 = 0 , 1 5 , 2 5 , 3 5 , 4 5 Hence, f ( x ) = 0 , when x = 5 n , n ∈ N = 1 when x = 5 n + 1 , n ∈ N = 2 when x = 5 n + 2 , n ∈ N = 3 when x = 5 n + 3 , n ∈ N = 4 when x = 5 n + 4 , n ∈ N Hence, f x is neither one-one nor onto.

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