Question

graph the following function on the axes provided.
$f(x) = \

$$\begin{cases} -3x - 9 & \\text{for} & -4 < x \\leq -1 \\\\ -x - 3 & \\text{for} & -1 < x \\leq 5 \\end{cases}$$

$

Explanation:

Step1: Analyze the first piece $f(x) = -3x - 9$ for $-4 < x \leq -1$

• Find two points on this line.
• When $x = -4$ (not included, open circle), $f(-4) = -3(-4) - 9 = 12 - 9 = 3$. So the point is $(-4, 3)$ with an open circle.
• When $x = -1$ (included, closed circle), $f(-1) = -3(-1) - 9 = 3 - 9 = -6$. So the point is $(-1, -6)$ with a closed circle.
• Plot these two points and draw a line segment between them.

Step2: Analyze the second piece $f(x) = -x - 3$ for $-1 < x \leq 5$

• Find two points on this line.
• When $x = -1$ (not included, open circle), $f(-1) = -(-1) - 3 = 1 - 3 = -2$. So the point is $(-1, -2)$ with an open circle.
• When $x = 5$ (included, closed circle), $f(5) = -5 - 3 = -8$. So the point is $(5, -8)$ with a closed circle.
• Plot these two points and draw a line segment between them.

Answer:

To graph the piecewise function:

1. For $f(x) = -3x - 9$ ($-4 < x \leq -1$):

• Plot an open circle at $(-4, 3)$ and a closed circle at $(-1, -6)$. Draw a line segment connecting these two points.

1. For $f(x) = -x - 3$ ($-1 < x \leq 5$):

• Plot an open circle at $(-1, -2)$ and a closed circle at $(5, -8)$. Draw a line segment connecting these two points.

(Note: Since this is a graphing problem, the final answer is the graphical representation as described above. If you were to sketch it on paper or using a graphing tool, follow these steps to plot the segments with the correct open/closed circles.)