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Graph the piecewise function f(x)={x+5 if x≤−2 2x+3 if x>−2

. Will the graph have any open endpoints? If yes, where?(1 point)
Responses

Yes, at (−2,3)
and (−2,−1)
.
Yes, at left parenthesis negative 2 comma 3 right parenthesis and left parenthesis negative 2 comma negative 1 right parenthesis .

No, all endpoints are closed.
No, all endpoints are closed.

Yes, at (−2,−1)
.
Yes, at left parenthesis negative 2 comma negative 1 right parenthesis .

Yes, at (−2,3)
.
Yes, at left parenthesis negative 2 comma 3 right parenthesis .

1 answer

To analyze the piecewise function ( f(x) = \begin{cases} x + 5 & \text{if } x \leq -2 \ 2x + 3 & \text{if } x > -2 \end{cases} ), let's first evaluate the function at \( x = -2 \):

  1. For \( x \leq -2 \): \[ f(-2) = -2 + 5 = 3 \] This means at \( x = -2 \), the point \( (-2, 3) \) is included in the graph.

  2. For \( x > -2 \): \[ f(-2) \text{ is not defined since we are only considering values greater than -2.} \] As a result, the point corresponding to \( x = -2 \) for the rule \( 2x + 3 \) will not be included, since it requires \( x \) to be strictly greater than -2.

Conclusion:

  • The graph will have an open endpoint at \( (-2, 3) \) from the first piece, since when we approach from the left, it is included, but from the piece for \( x > -2 \) (the second piece), it is not included at that point.
  • There will not be an open endpoint at \( (-2, -1) \) since that value isn't derived from either segment of the piecewise function.

Thus, the correct response is: Yes, at (−2, 3).