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

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

1 answer

Based on the description you provided regarding the two rays plotted on the coordinate plane, we can analyze the endpoints and behaviors of the rays to determine the correct piecewise function.

  1. Ray starting from an open endpoint: This signifies that the endpoint does not include that specific value in the function. Typically this indicates a type of inequality that does not include that value (e.g., > or <).

  2. Ray starting from a closed endpoint: This indicates that the endpoint is included within the function. This typically means the inequality used is inclusive (e.g., ≥ or ≤).

From the descriptions of the responses:

  • The first function, \( y = {-2, x > 0; 3, x > 0} \), has both parts only defined for \( x > 0 \), which doesn't differentiate between the open and closed behavior described.
  • The second function, \( y = {-2, x < 0; 3, x ≤ 0} \), incorrectly uses \( x < 0\) and \( x ≤ 0\), which are not appropriate for the described endpoints.
  • The third function, \( y = {-2x; 3x} \), does not provide the necessary piecewise distinction based on the endpoints.
  • The fourth function, \( y = {-2, x > 2; 3, x ≤ 3} \), suggests a closed endpoint at \( x = 3\), and may have a ray starting from \( x = 2\).

Based on your description stating that the first ray starts from an open endpoint and the second from a closed endpoint, I would suggest looking for a piecewise function that captures this behavior.

However, none of these options clearly reflect both characteristics. Could you confirm or clarify details or provide more context or numbers regarding where each ray starts?