Let $\mathbb{N}$ denote the set of all natural numbers, and $\mathbb{Z}$ denote the set of all integers. Consider the functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ and $g: \mathbb{Z} \rightarrow \mathbb{N}$ defined by \[ f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd }, \\ (4-n) / 2 & \text { if } n \text { is even },\end{cases} \] and \[ g(n)=\left\{\begin{array}{cl} 3+2 n & \text { if } n \geq 0, \\ -2 n & \text { if } n<0 . \end{array}\right. \] Define $(g \circ f)(n)=g(f(n))$ for all $n \in \mathbb{N}$, and $(f \circ g)(n)=f(g(n))$ for all $n \in \mathbb{Z}$. Then which of the following statements is (are) TRUE ?

(A) $g_\circ f$ is NOT one-one and $g \circ f$ is NOT onto; (D) $f$ is NOT one-one but $f$ is onto

(B) $f \circ \mathrm{~g}$ is NOT one-one but $f \circ g$ is onto

(C) $g$ is one-one and $g$ is onto

JEE Advanced · 2025 · 18 May 2025 · Paper 1 · Relations And Functions XII

Let

N

denote the set of all natural numbers, and

Z

denote the set of all integers. Consider the functions

f : N \to Z

and

g : Z \to N

defined by

f (n) = {(n + 1) /2 (4 - n) /2 if n is odd, if n is even,

and

g (n) = {3 + 2 n - 2 n if n \geq 0, if n < 0.

Define

(g \circ f) (n) = g (f (n))

for all

n \in N

, and

(f \circ g) (n) = f (g (n))

for all

n \in Z

.
Then which of the following statements is (are) TRUE ?

Answer: (A), (D)

Analyzing

f

f (1) = 1, f (2) = 1, f (3) = 2, f (4) = 0, \dots

. Since

f (1) = f (2)

f

is many-one. The range of

f

Z

, so

f

is onto. (D) is true.

Analyzing

g

: For

n \geq 0

g (n) = 3 + 2 n

gives odd values

\geq 3

. For

n < 0

g (n) = - 2 n

gives positive even values. Since these ranges don't overlap and cover distinct values,

g

is one-one. However,

g

misses 1 and 2, so it's not onto. (C) is false.

For

f \circ g

f (g (n)) = 2 + n

for

n \in N

, which is one-one and into (not onto

Z

). (B) is false.

For

g \circ f

g (f (1)) = g (1) = 5

and

g (f (2)) = g (1) = 5

, so

g \circ f

is many-one. It's also into (not onto

N

). (A) is true.

JEE Advanced · 2025 · 18 May 2025 · Paper 1 · One or more correct options