https://file.aiquickdraw.com/mathbot/file/1726604055486b4vdfi1d.jpg

Let's analyze the problem step by step.

### The function given is a piecewise function:
\[
f(x) = 
\begin{cases} 
-4x & \text{if } x < 0 \\
4x & \text{if } x \geq 0 
\end{cases}
\]

### (a) Graphing the function:
- For \(x < 0\), the function is \(f(x) = -4x\), which is a straight line with a negative slope, passing through the origin.
- For \(x \geq 0\), the function is \(f(x) = 4x\), which is a straight line with a positive slope, also passing through the origin.
- Therefore, the graph of this function consists of two lines meeting at the origin, one with a negative slope for negative \(x\), and one with a positive slope for positive \(x\).

In the image, option A shows this exact behavior:
- A line with a negative slope on the left side (for \(x < 0\)),
- A line with a positive slope on the right side (for \(x \geq 0\)).

### (b) Finding the range:
The range refers to all possible \(y\)-values of the function. 
- As \(x\) approaches negative infinity, \(y = -4x\) grows positively without bound.
- As \(x\) approaches positive infinity, \(y = 4x\) also grows positively without bound.
- Therefore, the range of the function is all real numbers \((- \infty, \infty)\).

Based on this analysis, the correct choice for the range is:
- **A. The range does not have any isolated values. It can be described by \((- \infty, \infty)\)**.

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Would you like a more detailed explanation or have any further questions?

### Related Questions:
1. What is the domain of a piecewise function?
2. How do you determine the slope of a line in a piecewise function?
3. What are common shapes for the graph of linear piecewise functions?
4. How does the behavior of the function at the boundary \(x = 0\) affect the graph?
5. How do you describe the range of a function that includes only certain values?

### Tip:
When graphing piecewise functions, always check for continuity at the boundary points to see if the function changes smoothly.

The domain of the piecewise function is (-∞, ∞). Graph the function and determine the function's range based on the graph.

Learn how to graph a piecewise function with linear components and determine its range. The function f(x) = -4x for x < 0 and f(x) = 4x for x >= 0 is graphed, and the range of the function is analyzed. Suitable for students in Grades 9-12.

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