JEE MAIN - Mathematics (2022 - 28th June Morning Shift - No. 1)
Let a function f : N $$\to$$ N be defined by
$$f(n) = \left[ {\matrix{ {2n,} & {n = 2,4,6,8,......} \cr {n - 1,} & {n = 3,7,11,15,......} \cr {{{n + 1} \over 2},} & {n = 1,5,9,13,......} \cr } } \right.$$
then, f is
one-one but not onto
onto but not one-one
neither one-one nor onto
one-one and onto
Explanation
When n = 1, 5, 9, 13 then $${{n + 1} \over 2}$$ will give all odd numbers.
When n = 3, 7, 11, 15 .....
n $$-$$ 1 will be even but not divisible by 4
When n = 2, 4, 6, 8 .....
Then 2n will give all multiples of 4
So range will be N.
And no two values of n give same y, so function is one-one and onto.
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