Question

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)
Responses

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

No, all endpoints are closed.

Answer 1

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

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

First, we need to evaluate the function at the point where the two pieces meet, which is \( x = -2 \).

1. For \( x \leq -2 \):
\[
f(-2) = -2 + 5 = 3
\]

2. For \( x > -2 \):
\[
f(-2) \text{ is not defined for this piece since we only consider values greater than } -2.
\]

Now, we evaluate if there are open endpoints:

- At \( x = -2 \), when approaching from the left side (values \( x \leq -2 \)), the function value is \( f(-2) = 3 \).
- When approaching from the right side (values \( x > -2 \)), there is no function value defined at \( x = -2 \).

Thus, since there is an endpoint at \( (-2, 3) \) that is included (closed), and no corresponding point defined for \( x = -2 \) as per the second piece, there is indeed an open endpoint at \( (-2, 3) \).

Therefore, the answer is:
**Yes, at \( (-2, 3) \)**.