https://file.aiquickdraw.com/mathbot/file/1727458561749zth0jvce.jpg

Let's analyze the piecewise-defined function provided in the image:

\[
f(x) = 
\begin{cases} 
\frac{1}{2}x, & \text{for } x < 0 \\
x + 3, & \text{for } x \geq 0
\end{cases}
\]

### 1. **Domain of the function**:
- The first piece, \(\frac{1}{2}x\), is valid for \(x < 0\).
- The second piece, \(x + 3\), is valid for \(x \geq 0\).
- Therefore, the domain of \(f(x)\) is all real numbers, \((-\infty, \infty)\), since there is no restriction on the values \(x\) can take. This matches option **B**: \((-\infty, \infty)\).

### 2. **Range of the function**:
- For \(x < 0\), \(\frac{1}{2}x\) produces values from \((-\infty, 0)\).
- For \(x \geq 0\), \(x + 3\) produces values starting from \(3\) and increasing to \((\infty)\).
- Combining these two parts, the range is \((-\infty, 3) \cup [3, \infty)\). This matches option **A**: \((-\infty, 0)\) and \([3, \infty)\).

### Conclusion:
- **Domain**: Option **B**: \((-\infty, \infty)\)
- **Range**: Option **A**: \((-\infty, 0)\) and \([3, \infty)\)

Would you like more details on any part of this explanation or have any questions?

Here are some related questions:
1. What is the general approach to finding the domain of piecewise functions?
2. How can we determine the range of a function from its equation?
3. Why is the range split into two parts in this case?
4. How does the slope of \(\frac{1}{2}x\) affect the function for \(x < 0\)?
5. How does the value of \(x+3\) behave for larger values of \(x\)?

**Tip**: When dealing with piecewise functions, carefully analyze each segment separately before combining the results for the domain and range.

Find the domain and the range of the function:
 f(x) = {1/2 x, for x < 0; x + 3, for x >= 0}

Learn how to find the domain and range of a piecewise-defined function with two parts: f(x) = 1/2 x for x < 0 and f(x) = x + 3 for x >= 0. This problem demonstrates key concepts in algebra and functions, suitable for high school students.

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