Graph the piecewise function f(x)={x+5 if x≤−22x+3 if x>−2. Will the graph have any open endpoints? If yes, where?(1 point)

Question

Responses

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

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

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 .

Yes, at (−2,−1).

Answer 1

To analyze the piecewise function \( f(x) \):

1. Function Definition:

   • For \( x \leq -2 \), \( f(x) = x + 5 \)
   • For \( x > -2 \), \( f(x) = 2x + 3 \)

2. Finding the function values at \( x = -2 \):

   • For \( x = -2 \):
     • If \( x \leq -2 \), \( f(-2) = -2 + 5 = 3 \)
     • If \( x > -2 \), \( f(-2) \) is not defined.

3. Endpoints:

   • At \( x = -2 \), the function has a value \( f(-2) = 3 \) (from the first piece).
   • The next piece, \( f(x) = 2x + 3 \), will not include the value at \( x = -2 \) since it only applies for \( x > -2 \).

Therefore, there is an open endpoint at \( (-2, 3) \) because while it is included in the first piece, the second piece does not include this point.

Conclusion

Yes, there is an open endpoint at \( (-2, 3) \).

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