Graph the piecewise function f(x)={x+5 if x≤−22x+3 if x>−2

Question

𝑓
(
𝑥
)
=
{
𝑥
+
5

if

𝑥
≤
−
2
2
𝑥
+
3

if

𝑥
>
−
2
. Will the graph have any open endpoints? If yes, where?(1 point)
Responses

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

Yes, 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,−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

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

we need to evaluate the function at the breakpoint, which is \( x = -2 \).

1. For the first piece of the function where \( x \leq -2 \): \[ f(-2) = (-2) + 5 = 3 \]

2. For the second piece of the function where \( x > -2 \): \[ f(-2) \text{ is not included in this piece.} \]

Now, we determine if there are open endpoints:

• At \( x = -2 \), the value from the first part is \( f(-2) = 3 \), which is included (closed).
• The second part does not define \( f(-2) \) since it only applies for \( x > -2 \).

Since the function does not have a value defined for \( x = -2 \) in the second piece, we have an open endpoint at the point \( (-2, 3) \).

Conclusion: Yes, the graph has an open endpoint at \( (-2, 3) \).

So the correct response is: Yes, at (-2, 3).