To understand the piecewise function given as:
\[ F(x) = \begin{cases} x + 3 & \text{if } x < -2 \ 2x + 3 & \text{if } x > -2 \end{cases} \]
we first need to analyze the behavior of the function around the point \(x = -2\).
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For \(x < -2\): The function is defined as \(F(x) = x + 3\). If we evaluate this at \(x = -2\): \[ F(-2) = -2 + 3 = 1 \] However, this value is not included in the function because the condition states \(x < -2\).
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For \(x > -2\): The function is defined as \(F(x) = 2x + 3\). We do not evaluate this at \(x = -2\) since the condition is strictly greater than -2.
However, we can find the limit as \(x\) approaches -2 from the right: \[ \lim_{x \to -2^+} F(x) = 2(-2) + 3 = -4 + 3 = -1 \]
Since the value of the function at \(x = -2\) does not exist (the function does not include \(x = -2\)) and has different limits approaching from the left and the right, we conclude that there will be an open endpoint at \((-2, 1)\) for \(x < -2\) and a value of \((-2, -1)\) from the right.
Thus, the graph of the function will have an open endpoint at:
- (-2, 1) from the left at \(x < -2\) (since it is not included)
- (-2, -1) from the right at \(x > -2\) (since it is also not included)
Now looking at the answer choices provided:
- No, all endpoints are closed.
- Yes, at (-2, 3)
- Yes, at (-2, -1)
- Yes, at (-2, 3) and (-2, -1)
The correct answer is:
3. Yes, at (-2, -1)
for the endpoint from the right. But since the graph does not touch or include either (-2, 1) or (-2, -1), you could argue that the correct overall endpoint notation might be faint from the choices and the situation.
In any case, since the prompt asks directly about an open endpoint, it can be considered that you have two open ends; however, since the options don't overtly provide the (-2, 1) option, then selecting 3. Yes, at (-2, -1) is the strongest response.