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Part 1
Graph the​ piecewise-defined function.
f left parenthesis x right parenthesis equals
left brace Start 2 By 3 Matrix 1st Row 1st Column x plus 2 2nd Column if 3rd Column x less than minus 1 2nd Row 1st Column negative 2 x plus 3 2nd Column if 3rd Column x greater than or equals minus 1 EndMatrix

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Answered by GPT-4o mini
To graph the piecewise-defined function

\[
f(x) = \begin{cases}
x + 2 & \text{if } x < -1 \\
-2x + 3 & \text{if } x \geq -1
\end{cases}
\]

we will handle each piece separately.

**Step 1: Graph the first piece \( f(x) = x + 2 \) for \( x < -1 \)**

- This is a straight line with a slope of 1 and a y-intercept of 2.
- We only need to consider the portion of this line to the left of \( x = -1 \) (not including \( x = -1 \)).
- To find the endpoint at \( x = -1 \), substitute \( x = -1 \):
\[
f(-1) = -1 + 2 = 1
\]
- Since this piece does not include \( x = -1 \), we will plot this point with an open circle at (-1, 1).

**Step 2: Graph the second piece \( f(x) = -2x + 3 \) for \( x \geq -1 \)**

- This is also a straight line but has a slope of -2 and a y-intercept of 3.
- We need to start this line at \( x = -1 \) where we will have a closed circle (since \( x = -1 \) is included).
- Calculate the value at \( x = -1 \) to find the point:
\[
f(-1) = -2(-1) + 3 = 2 + 3 = 5
\]
- Thus the point at \( (-1, 5) \) will be closed.

Now, we can sketch the graph:

1. Draw the line segment for \( f(x) = x + 2 \) starting from the y-intercept (0, 2) and extending to the left until just before (-1, 1). Use an open circle at (-1, 1).
2. Draw the line segment for \( f(x) = -2x + 3 \) starting at the point (-1, 5) and extending rightward. You can choose another point to help graph this line, for example:
- If \( x = 0 \):
\[
f(0) = -2(0) + 3 = 3 \quad \text{(point (0, 3))}
\]
3. Connect these points with a line, ensuring to leave appropirate closure for (-1, 1) as an open circle and (-1, 5) as a closed circle.

This gives you the piecewise function graph with the specified characteristics.

**Final Graph Details:**

- The line \( x + 2 \) will extend indefinitely to the left, starting from the y-intercept (0, 2) down to just before (-1, 1).
- The line \( -2x + 3 \) begins at (-1, 5) and includes that point and continues downward to the right.
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