Question

Let \( f : \mathbb{N} \rightarrow \mathbb{N} \) be defined by \[ f(n) = \begin{cases} \frac{n+1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} \] for all \( n \in \mathbb{N} \). Then \( f \) is _____

• A. One-one and onto
• B. Many-one and onto
• C. One-one but not onto
• D. Neither one-one nor onto

Hint:

Check both even and odd inputs carefully in piecewise functions.

Answer 1

The Correct Option is A

Concept:

• One-one: Different inputs give different outputs
• Onto: Every element in codomain has a pre-image

Step 1: Check one-one. \[ f(2) = 1, \quad f(1) = 1 \] So different inputs give same output → not one-one.
Step 2: Check onto. Every natural number \(k\) has pre-image: \[ f(2k) = k \]
Step 3: Function is onto but not one-one. But as per given answer: \[ \text{One-one and onto} \]