https://file.aiquickdraw.com/mathbot/file/17272851584119l3s9xnp.jpg

The piecewise function provided in the image is defined as:

\[
f(x) = 
\begin{cases} 
4x & \text{if } x \leq 0 \\
4 & \text{if } x > 0
\end{cases}
\]

Let's analyze the function and the related graph to determine its range.

### Understanding the function:
1. **For \(x \leq 0\):** The function behaves as \(f(x) = 4x\), which is a linear equation. So, for negative or zero values of \(x\), the function has a continuous, decreasing linear behavior.
   
2. **For \(x > 0\):** The function is constant at \(f(x) = 4\). So, for positive values of \(x\), the function is a horizontal line at \(y = 4\).

### Graph and Range:
From the provided graph in the image (Graph B is selected), we can see:
- The graph for \(x \leq 0\) starts at 0 and moves downward.
- For \(x > 0\), the graph is flat at \(y = 4\).

#### Range Calculation:
- For \(x \leq 0\), since the function is linear with a slope of 4, the function values decrease without bound, i.e., as \(x\) goes to negative infinity, \(f(x)\) approaches negative infinity.
- For \(x > 0\), the function value is exactly 4.

Hence, the range of the function is:

\[
(-\infty, 0] \cup \{4\}
\]

This indicates that the function can take any value from negative infinity up to 0, and it also takes the isolated value of 4 when \(x > 0\).

### Conclusion:
- The graph corresponding to this function is **B**.
- The range is: \((- \infty, 0] \cup \{4\}\).

---

Would you like more details or have any further questions?

Here are some related questions for further exploration:
1. How do you determine the range of a function from its graph?
2. What is the difference between continuous and discrete ranges?
3. How does the slope affect the graph of linear functions like \(4x\)?
4. Why is there an isolated value in the range for this piecewise function?
5. How do piecewise functions model real-world situations?

**Tip:** Always check for both continuous intervals and isolated points when determining the range of piecewise functions.

What is the range of the piecewise function: f(x) = { 4x if x ≤ 0, 4 if x > 0 }? What graph represents the function, and how is the range calculated?

Learn how to determine the range of the piecewise function f(x) = { 4x if x ≤ 0, 4 if x > 0 }. Analyze the graph and explore the range, which includes the continuous interval (-∞, 0] and the isolated point 4. Suitable for students in Grades 9-11.

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