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

Yes, at (−2,−1)
.

Answer 1

To evaluate whether the graph of the piecewise function has open endpoints, we need to analyze the function given.

The function is defined as follows:

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

Let's examine the endpoints:


1. For \( x = -2 \):
- The first piece gives \( f(-2) = -2 + 5 = 3 \). This is a closed endpoint because \( x = -2 \) is included.
- The second piece, \( 2x + 3 \), does not apply at \( x = -2 \) since it's defined only for \( x > -2 \).

2. The second piece \( 2x + 3 \) starts just to the right of \( x = -2 \). At \( x = -2 \) (from the second piece), \( f(-2) \) is not defined, implying an open endpoint here.

Given this analysis, we conclude:

- The graph has open endpoints precisely at \( (-2, 3) \) from the first piece (inclusive) and \( (-2, -1) \) where the second piece starts (exclusive).
- So the only open endpoint is at \( (-2, 3) \).

### Answer
Yes, at \((-2, 3)\).