Ask a New Question
Menu

Graphthe 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)

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 .

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

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

Yes, at (−2,3).

1 answer

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

\[ f(x) = \begin{cases} x + 5 & \text{if } x \leq -2 \ 2x + 3 & \text{if } x > -2 \end{cases} \]

1. Finding the y-values at the endpoint \( x = -2 \):

  • For \( x \leq -2 \) (specifically at \( x = -2 \)): \[ f(-2) = -2 + 5 = 3 \]

  • For \( x > -2 \) (specifically at \( x = -2 \)): \[ \text{This piece of the function does not include } x = -2. \] However, we can find \( f(x) \) approaching \( -2 \) from the right: \[ \lim_{x \to -2^+} f(x) = 2(-2) + 3 = -4 + 3 = -1 \]

2. Analyzing the endpoints:

  • At \( x = -2\):
    • The function value is \( 3 \) (from the left side).
    • The limit from the right approaches \( -1 \).

This means that the graph will have an open endpoint at the point \( (-2, -1) \), because the value at \( x = -2 \) does not include \( -2 \) in the second piece of the piecewise function (which only includes values greater than \( -2 \)).

Conclusion:

Answer: Yes, at \( (-2, 3) \) and \( (-2, -1) \).

Final Response:

Yes, at (−2,3) and (−2,−1).