If `f: N to N ` defined as `f(x)=x^(2)AA x in N, ` then f is :

Options

A. many-one
B. one-one
C. onto
D. none of these

Answer

To determine the nature of the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x^2 \), we will analyze whether the function is one-one (injective) and onto (surjective).

### Step 1: Check if \( f \) is one-one (injective)
To check if \( f \) is one-one, we need to see if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).

1. Assume \( f(x_1) = f(x_2) \).
2. This gives us \( x_1^2 = x_2^2 \).
3. Taking the square root of both sides, we have \( x_1 = \pm x_2 \).
4. Since \( x_1 \) and \( x_2 \) are both in \( \mathbb{N} \) (natural numbers), the negative solution is not valid. Therefore, we conclude that \( x_1 = x_2 \).

Thus, \( f \) is one-one.

### Step 2: Check if \( f \) is onto (surjective)
To check if \( f \) is onto, we need to see if for every \( y \in \mathbb{N} \), there exists an \( x \in \mathbb{N} \) such that \( f(x) = y \).

1. Let \( y \) be any natural number.
2. We want to find \( x \) such that \( f(x) = y \), which means \( x^2 = y \).
3. Solving for \( x \), we get \( x = \sqrt{y} \).
4. For \( x \) to be a natural number, \( y \) must be a perfect square (e.g., 1, 4, 9, 16, etc.). However, not all natural numbers are perfect squares.

Since there are natural numbers (like 2, 3, 5, etc.) that cannot be expressed as the square of another natural number, \( f \) is not onto.

### Conclusion
The function \( f(x) = x^2 \) is one-one but not onto.

### Final Answer
The function \( f \) is one-one but not onto.