Let f : `N rarr N` be defined as f(x) = 2x for all `x in N`, then f is

Options

A. Onto
B. Invertible
C. One-one
D. Many one

Answer

To determine the properties of the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = 2x \) for all \( x \in \mathbb{N} \), we will analyze whether the function is one-to-one (injective), onto (surjective), and if it is invertible.

### Step-by-step Solution:

1. **Understanding the Function**:
The function is defined as \( f(x) = 2x \). This means for every natural number \( x \), the output is double that number.

2. **Checking if the Function is One-to-One (Injective)**:
A function is one-to-one if different inputs produce different outputs.
- Let's assume \( f(a) = f(b) \) for some \( a, b \in \mathbb{N} \).
- This means \( 2a = 2b \).
- Dividing both sides by 2 gives us \( a = b \).
- Since \( a \) must equal \( b \), the function is injective.

3. **Checking if the Function is Onto (Surjective)**:
A function is onto if every element in the codomain (range) has a pre-image in the domain.
- The codomain here is \( \mathbb{N} \).
- The outputs of the function \( f(x) = 2x \) are all even natural numbers (2, 4, 6, ...).
- However, the odd natural numbers (1, 3, 5, ...) do not have pre-images in \( \mathbb{N} \) because there is no natural number \( x \) such that \( f(x) \) would yield an odd number.
- Therefore, the function is not onto.

4. **Conclusion about Invertibility**:
A function is invertible if it is both one-to-one and onto.
- Since \( f \) is one-to-one but not onto, it cannot be invertible.

5. **Final Classification**:
Based on the analysis, we can conclude:
- The function \( f(x) = 2x \) is one-to-one (injective) but not onto (surjective) and therefore not invertible.

### Summary of Properties:
- **One-to-One (Injective)**: Yes
- **Onto (Surjective)**: No
- **Invertible**: No