Question

Graph the piecewise function f(x)={x+5 if x≤−22x+3 if x>−2𝑓(𝑥)={𝑥+5 if 𝑥≤−22𝑥+3 if 𝑥>−2 . Will the graph have any open endpoints? If yes, where?(1 point)ResponsesYes, at (−2,3)(−2,3) and (−2,−1)(−2,−1).Yes, at left parenthesis negative 2 comma 3 right parenthesis and left parenthesis negative 2 comma negative 1 right parenthesis .Yes, at (−2,3)(−2,3).Yes, at left parenthesis negative 2 comma 3 right parenthesis .Yes, at (−2,−1)(−2,−1).Yes, at left parenthesis negative 2 comma negative 1 right parenthesis .No, all endpoints are closed.

Answer 1

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. For \( x \leq -2 \), the function is \( f(x) = x + 5 \).
- At \( x = -2 \):
\[
f(-2) = -2 + 5 = 3
\]
- This point is closed because it includes \( -2 \) in the domain.

2. For \( x > -2 \), the function is \( f(x) = 2x + 3 \).
- The endpoint at \( x = -2 \) does not fall in this interval (since it is strictly greater).
- We can find the value of the function as \( x \) approaches -2:
\[
\text{As } x \to -2^+, f(x) = 2(-2) + 3 = -4 + 3 = -1
\]
- This point is open because it excludes \( -2 \).

### Conclusion on open endpoints:
- There will be an open endpoint at the value \( f(-2) \) from the second case, which is \( (-2, -1) \).
- There will also be a closed point at \( (-2, 3) \) from the first case.

### Answer:
Yes, there will be open endpoints at \( (-2, -1) \).