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Functions question

2025 · Shift 1 · Q22

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PYQ Library
JEE Advanced
Mathematics
Functions
2025 · Shift 1 · Q22

Functions question

2025 · Shift 1 · Q22

JEE AdvancedMathematicsFunctionsMultiple correct+4 / −2

Let ℕ denote the set of all natural numbers, and ℤ denote the set of all integers. Consider the functions f: ℕ → ℤ and g: ℤ → ℕ defined by

f(n) = \begin{cases} \frac{(n + 1)}{2} & \text{if } n \text{ is odd,} \\ \frac{(4-n)}{2} & \text{if } n \text{ is even,} \end{cases}

and

g(n) = \begin{cases} 3 + 2n & \text{if } n \ge 0 , \\ -2n & \text{if } n \lt 0 . \end{cases}

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
B
f $\circ$ g is NOT one-one but f $\circ$ g is onto
C
g is one-one and g is onto
D
f is NOT one-one but f is onto

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