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

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 .

No, all endpoints are closed.

Answer 1

To determine if the piecewise function has any open endpoints, we need to analyze the two parts of the function:

1. The first piece is \( f(x) = x + 5 \) for \( x \leq -2 \).
2. The second piece is \( f(x) = 2x + 3 \) for \( x > -2 \).

Next, we evaluate the function at the point \( x = -2 \):

- For \( x = -2 \) under the first piece:
\[
f(-2) = -2 + 5 = 3
\]

- For \( x = -2 \) under the second piece:
This piece is not applicable since it is defined only for \( x > -2 \).

This indicates that the point \( (-2, 3) \) is defined by the first piece, making it a closed endpoint for the first piece of the function.

Now we check \( f(x) \) as \( x \) approaches \(-2\) from the right using the second piece:
- As \( x \) approaches \(-2\) from the right, we calculate:
\[
f(x) = 2x + 3 \text{ at } x = -2 \rightarrow f(-2) = 2(-2) + 3 = -4 + 3 = -1.
\]

Even though we can evaluate the function very close to \(-2\) using the second piece, it is technically not defined at \( x = -2 \) for \( f(x) = 2x + 3 \), which results in an open endpoint.

Thus, the endpoints at \( x = -2 \) are:
- Closed at \( ( -2, 3 ) \)
- Open at \( ( -2, -1 ) \)

So yes, the graph will have open endpoints. The correct response is:

**Yes, at \((-2, -1)\).**