Question

Use the image to answer the question.

A coordinate plane's axes range from negative 5 to 5 by 1-unit increments. Two rays are plotted. The first ray starts from a closed endpoint, while the second ray starts from an open endpoint.

What is the piecewise function for the graph?

Option #1:

f(x)={−2x+1, x<023x−3, x≥0

Option #2:

f(x)={5, x≤22x−4, x>2

(1 point)
Option #
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Answer 1

To create a piecewise function based on the description provided, let's analyze the two options given:

**Option #1:**
\[ f(x) =
\begin{cases}
-2x + 1 & \text{if } x < 0 \\
3x - 3 & \text{if } x \geq 0
\end{cases}
\]

**Option #2:**
\[ f(x) =
\begin{cases}
5 & \text{if } x \leq 2 \\
2x - 4 & \text{if } x > 2
\end{cases}
\]

### Evaluating the Options

Given the description of the rays from the image (assuming we want to identify which piecewise function corresponds to them based on their characteristics):

1. **First Ray:**
- Starts from a closed endpoint, indicating that the function includes that point. This usually corresponds to the condition involving \( x \geq \) or \( x \leq \).

2. **Second Ray:**
- Starts from an open endpoint, meaning that the function does not include that endpoint. This generally corresponds to the condition involving strict inequalities, such as \( x < \) or \( x > \).

### Analyzing the Options in Context

- In **Option #1**, the first piece applies when \( x < 0 \) (likely an open ray as it’s less than 0) and the second piece applies when \( x \geq 0\) (a closed ray starting at 0).
- In **Option #2**, the first piece is constant at \( 5 \) for \( x \leq 2\) (closed endpoint at 2) and the second piece \( 2x - 4\) for \( x > 2\) (open ray starting at 2).

### Conclusion

Without the image, it is difficult to definitively declare which piecewise function is correct, but the structure of these options aligns with the endpoint descriptions as follows:

- If the closed endpoint is at \(0\) and the rays continue from there, **Option #1** would correspond to that.
- If the closed endpoint is at \(2\) and the rays continue from there, then **Option #2** fits.

Therefore, I would say:

**Choose Option #1 if the closed endpoint is at \(0\)**, or **choose Option #2 if the closed endpoint is at \(2\)**.