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

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

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 (−2,3)
 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)
.

GPT-4o mini · Answer

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 look at the two cases defined by the function: 1. For $ x \leq -2 $, the function is defined as $ f(x) = x + 5 $. 2. For $ x > -2 $, the function is defined as $ f(x) = 2x + 3 $. Now let's evaluate the function at the point $ x = -2 $: - In the first case when $ x = -2 $: $$ f(-2) = -2 + 5 = 3 $$ - In the second case, where the condition is $ x > -2 $, we also need to check the limit of $ f(x) $ as $ x $ approaches $-2$: $$ \lim_{x \to -2^+} f(x) = 2(-2) + 3 = -4 + 3 = -1 $$ At $ x = -2 $: - From the left ($ x \leq -2 $), the function value is $ 3 $. - From the right ($ x > -2 $), the limit approaches $ -1 $. Since the value at $ x = -2 $ is $ 3 $ and the limit from the right does not equal this value, there will be an open endpoint at $ (-2, 3) $. Thus, the graph will have an open endpoint at $ (-2, 3) $ because while the value exists at that point for $ x \leq -2 $, it is not included in the function for $ x > -2 $. Final answer: **Yes, at (-2, 3)**.