Let N denote the set of all non-negative integers and Z denote the set of all integers. The function `f:ZtoN` given by `f(x)=|x|` is:

Options

A. One-one but not onto
B. Onto but not one-one
C. Both one-one and onto
D. Neither one-one nor onto

Explanation

To determine the properties of the function \( f: \mathbb{Z} \to \mathbb{N} \) defined by \( f(x) = |x| \), we need to analyze whether the function is one-to-one (injective) and/or onto (surjective).

### Step 1: Check if the function is one-to-one (injective)

A function is one-to-one if different inputs produce different outputs. In mathematical terms, \( f(a) = f(b) \) implies \( a = b \).

- Consider two different integers, \( a \) and \( b \), such that \( f(a) = f(b) \).
- This means \( |a| = |b| \).
- For example, if \( a = -1 \) and \( b = 1 \), we have \( f(-1) = | -1 | = 1 \) and \( f(1) = | 1 | = 1 \). Here, \( a \neq b \) but \( f(a) = f(b) \).

Since we can find distinct integers \( a \) and \( b \) such that \( f(a) = f(b) \), the function \( f \) is not one-to-one.

### Step 2: Check if the function is onto (surjective)

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

- The codomain \( \mathbb{N} \) consists of all non-negative integers: \( 0, 1, 2, 3, \ldots \).
- For any non-negative integer \( n \in \mathbb{N} \), we can find an integer \( x \in \mathbb{Z} \) such that \( f(x) = n \).
- Specifically, if we take \( x = n \) or \( x = -n \), we have \( f(n) = |n| = n \) and \( f(-n) = |-n| = n \).

Since every non-negative integer \( n \) can be achieved by some integer \( x \), the function \( f \) is onto.

### Conclusion

Based on the analysis:
- The function \( f \) is **not one-to-one** (injective).
- The function \( f \) is **onto** (surjective).

Thus, the correct answer is that the function is onto but not one-to-one.

### Summary of the Answer
The function \( f: \mathbb{Z} \to \mathbb{N} \) defined by \( f(x) = |x| \) is **onto but not one-to-one**.

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