Use the image to answer the question.

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>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={−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 less than zero line 2 3 comma x less than or equal to zero

y={−2, x>23, x≤3

Answer 1

To determine the correct piecewise function based on the description provided:

• The first ray starts from an open endpoint. This means that the value associated with this ray is not included in the piecewise function. If we denote this ray as resulting in a constant value of -2, it will be for values of \(x > 0\).

• The second ray starts from a closed endpoint. This suggests that the corresponding value (3) is included, applicable for \(x \geq 0\).

Given these interpretations, we can analyze the possible responses:

1. y={−2, x>03, x>0:

   • This means \(y = -2\) for \(x > 0\) and \(y = 3\) for \(x > 0\). This is inconsistent since both cannot be true.

2. y={−2x3x:

   • This suggests some form of multiplication, but it lacks clear separation of conditions. Therefore, it is not correctly formatted.

3. y={−2, x<03, x≥0:

   • This means \(y = -2\) for \(x < 0\) (which is consistent with the ray starting from the left side) and \(y = 3\) for \(x \geq 0\) (which is consistent with the ray originating from a closed point on the right). This interpretation fits the description well.

4. y={−2, x>23, x≤3:

   • This means \(y = -2\) for \(x > 2\) (which does not match the left ray description) and \(y = 3\) for \(x \leq 3\) (which is also too broad for the ray starting from the closed point).

Considering the above analysis, the correct piecewise function is:

y={−2, x<03, x≥0.