Prove that the function `f: N->N` , defined by `f(x)=x^2+x+1` is one-one but not onto.

Explanation

To prove that the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x^2 + x + 1 \) is one-one but not onto, we will follow these steps:

### Step 1: Prove that \( f \) is one-one (injective)

To show that \( f \) is one-one, we need to demonstrate that if \( f(x) = f(y) \), then \( x = y \) for all \( x, y \in \mathbb{N} \).

1. Assume \( f(x) = f(y) \).
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