Question

Graph the piecewise function f(x)={x+5 if x≤−2, 2x+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) 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 (−2,3) . Yes, at left parenthesis negative 2 comma 3 right parenthesis . No, all endpoints are closed.

Answer 1

To determine whether the graph of the piecewise function \( f(x) \) has any open endpoints, we need to analyze the two parts of the function:

1. For \( x \leq -2 \): \[ f(x) = x + 5 \] At \( x = -2 \): \[ f(-2) = -2 + 5 = 3 \] This means that at the point \( (-2, 3) \), there is a closed endpoint since this part of the function includes the endpoint at \( x = -2 \).

2. For \( x > -2 \): \[ f(x) = 2x + 3 \] The value of \( f(x) \) as \( x \) approaches \(-2\) from the right is: \[ \lim_{x \to -2^+} f(x) = 2(-2) + 3 = -4 + 3 = -1 \] Since this segment of the function does not include \( x = -2 \) (it only applies for values greater than \(-2\)), the point \( (-2, -1) \) will be an open endpoint.

Based on this analysis:

• The point \( (-2, 3) \) is where \( f(x) \) is defined for \( x \leq -2 \) (closed).
• The point \( (-2, -1) \) is where \( f(x) \) is defined for \( x > -2 \) (open).

Thus, the answer to the question of whether the graph will have any open endpoints is:

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