Let $f, g: \mathbb{N}-\{1\} \rightarrow \mathbb{N}$ be functions defined by $f(\mathrm{a})=\alpha$, where $\alpha$ is the maximum of the powers of those primes p such that $p^{\alpha}$ divides $a$, and $g(a)=a+1$, for all $a \in \mathbb{N}-\{1\}$. Then, the function $f+\mathrm{g}$ is

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Sets Relations and Functions

Let

f, g : N - {1} \to N

be functions defined by

f (a) = α

, where

α

is the maximum of the powers of those primes p such that

p^{α}

divides

a

, and

g (a) = a + 1

, for all

a \in N - {1}

. Then, the function

f + g

Answer: (D)

Computing values:

f (2) = 1, g (2) = 3

(f + g) (2) = 4

f (3) = 1, g (3) = 4

(f + g) (3) = 5

f (4) = 2, g (4) = 5

(f + g) (4) = 7

f (5) = 1, g (5) = 6

(f + g) (5) = 7

.

Since

(f + g) (4) = (f + g) (5) = 7

, the function is many-one (not one-one). Also,

(f + g) (x)

does not contain 1 in its range, so it is not onto. Answer: (D)

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option