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Question

A coordinate plane's axes range from negative 10 to 10 by 1-unit increments. Two rays are plotted. The first ray starts from an open endpoint, while the second ray starts from a closed endpoint.

Choose the correct piecewise function.

(1 point)
Responses

y={−2, x>23, x≤3
y equals left brace line 1 negative 2 comma x less than 2 line 2 3 comma x less than or equal to 3

y={−2x3x
y equals left brace line 1 negative 2 time x line 2 3 times x

y={−2, x>03, x>0
y equals left brace line 1 negative 2 comma x greater than 0 line 2 3 comma x less than 0

y={−2, x<03, x≥0
y equals left brace line 1 negative 2 comma x less than zero line 2 3 comma x less than or equal to zero
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Answer 1

To determine the correct piecewise function based on the description provided, we need to analyze the conditions for the rays and their endpoints.

Since the problem mentions:

• The first ray starts from an open endpoint, which indicates that the function does not include its value at a certain point (likely where the value changes).
• The second ray starts from a closed endpoint, indicating that the function does include its value at a certain point.

Given the available function options and the description of the rays, we can deduce the following:

1. Open endpoint: Typically signifies a boundary where the function changes but does not include that value. This usually represents a transition in the piecewise function.

2. Closed endpoint: Indicates the function value at that point is included.

Now, let's evaluate the function options:

1. y={−2, x > 2; 3, x ≤ 3
   This suggests that for \(x > 2\), \(y = -2\) (open endpoint), and for \(x \leq 3\), \(y = 3\) (closed endpoint).

2. y={−2x; 3x
   This format doesn't match the piecewise representation correctly and is likely incorrect.

3. y={−2, x > 0; 3, x < 0
   In this case, the endpoints imply the first ray starts from an open endpoint at \(x = 0\) (turning point) while the second ray (at \(y = 3\)) would go left. However, both intervals overlap suspiciously.

4. y={−2, x < 0; 3, x ≥ 0
   Here, \(y = -2\) applies to all \(x < 0\) which suggests the function approaches -2 but does not reach it at zero (open endpoint on the left). From \(0\) and onwards (\(x \geq 0\)), it becomes \(3\).

Based on the distinct descriptions of the endpoints and the ranges, the last option makes the most sense and aligns with the provided details.

Therefore, the correct piecewise function is:

y={−2, x < 0; 3, x ≥ 0