Give an example of a function which is one-one but not onto.

Explanation

To provide a step-by-step solution to the question, we will define a function that is one-one but not onto, and then explain why it meets these criteria.

### Step-by-Step Solution:

1. **Define the Function**:
We will define a function \( f: \mathbb{Z} \to \mathbb{R} \) by \( f(x) = x \).

2. **Check if the Function is One-One (Injective)**:
A function is one-one if different inputs map to different outputs.
- Let \( f(a) = f(b) \) for \( a, b \in \mathbb{Z} \).
- This implies \( a = b \).
- Hence, \( f \) is one-one.

3. **Determine the Range of the Function**:
The range of the function \( f \) is the set of all outputs it can produce.
- Since \( f(x) = x \) and \( x \) can take any integer value, the range of \( f \) is \( \mathbb{Z} \).

4. **Identify the Co-Domain**:
The co-domain of the function is the set of all possible outputs defined in the function's definition.
- Here, the co-domain is \( \mathbb{R} \).

5. **Check if the Function is Onto (Surjective)**:
A function is onto if every element in the co-domain has a pre-image in the domain.
- Since the range of \( f \) is \( \mathbb{Z} \) and the co-domain is \( \mathbb{R} \), there are real numbers (like \( \pi \), \( e \), etc.) that are not integers.
- Therefore, not every element in \( \mathbb{R} \) has a corresponding integer in \( \mathbb{Z} \).

6. **Conclusion**:
Since \( f \) is one-one but not onto, we can conclude that the function \( f: \mathbb{Z} \to \mathbb{R} \) defined by \( f(x) = x \) is an example of a function that is one-one but not onto.

### Final Answer:
The function \( f: \mathbb{Z} \to \mathbb{R} \) defined by \( f(x) = x \) is one-one but not onto.

---