Let a function $\mathrm{f}: \mathbb{N} \rightarrow \mathbb{N}$ be defined by $$f(\mathrm{n})=\left[\begin{array}{ll}2 \mathrm{n}, & \mathrm{n}=2,4,6,8, \ldots \ldots \\ \mathrm{n}-1, & \mathrm{n}=3,7,11,15, \ldots \ldots \\ \frac{\mathrm{n}+1}{2}, & \mathrm{n}=1,5,9,13, \ldots \ldots\end{array}\right.$$ then, f is

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Sets Relations and Functions

Let a function

f : N \to N

be defined by

f (n) = 2 n, n - 1, \frac{n + 1}{2}, n = 2, 4, 6, 8, \dots\dots n = 3, 7, 11, 15, \dots\dots n = 1, 5, 9, 13, \dots\dots

then, f is

Answer: (D)

Rewrite the function in terms of remainders modulo 4:

Every natural number falls into exactly one of these categories. The ranges are:

These three sets partition

N

, and for each output value, there is exactly one input value. Therefore, f is both one-one and onto.

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Single correct option