The function f N `rarr` N. where N being the set of natural numbers, defined by f(x) = 2x + 3 is:

Options

A. Injective and surjective
B. Injective but not surjective
C. Not injective hut surjective
D. Neither injective nor surjective

Answer

To determine the properties of the function \( f: \mathbb{N} \rightarrow \mathbb{N} \) defined by \( f(x) = 2x + 3 \), we need to analyze whether it is injective (one-to-one) and surjective (onto).

### Step 1: Check if the function is injective.

A function is injective if different inputs produce different outputs. To check this, we can assume \( f(x_1) = f(x_2) \) and see if it implies \( x_1 = x_2 \).

1. Assume \( f(x_1) = f(x_2) \).
2. This means \( 2x_1 + 3 = 2x_2 + 3 \).
3. Subtract 3 from both sides: \( 2x_1 = 2x_2 \).
4. Divide both sides by 2: \( x_1 = x_2 \).

Since \( x_1 = x_2 \) holds true, the function is injective.

### Step 2: Check if the function is surjective.

A function is surjective if every element in the codomain (in this case, \( \mathbb{N} \)) has a pre-image in the domain.

1. We need to check if for every \( y \in \mathbb{N} \), there exists an \( x \in \mathbb{N} \) such that \( f(x) = y \).
2. Set \( y = 2x + 3 \).
3. Rearranging gives \( 2x = y - 3 \).
4. Thus, \( x = \frac{y - 3}{2} \).

For \( x \) to be a natural number, \( y - 3 \) must be non-negative and even. This means \( y \) must be at least 3 and must be odd (since \( y - 3 \) must be divisible by 2).

Thus, the function is not surjective because not every natural number can be expressed in the form \( 2x + 3 \).

### Conclusion

The function \( f(x) = 2x + 3 \) is injective but not surjective.

### Final Answer
The function \( f: \mathbb{N} \rightarrow \mathbb{N} \) defined by \( f(x) = 2x + 3 \) is injective but not surjective.

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