Question

question
graph the following function on the axes provided.

$f(x) = \

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

$

click and drag to make a line. click the line to delete it.
click on an endpoint of a line to change it.

graph axes with x from -10 to 10 and y from -10 to 10

Explanation:

Step1: Analyze the first piece

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

• When \( x = -4 \) (not included, open circle), \( f(-4) = -3(-4) - 9 = 12 - 9 = 3 \).
• When \( x = -1 \) (included, closed circle), \( f(-1) = -3(-1) - 9 = 3 - 9 = -6 \).

Plot the open circle at \((-4, 3)\) and closed circle at \((-1, -6)\), then draw the line between them.

Step2: Analyze the second piece

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

• When \( x = -1 \) (not included, open circle), \( f(-1) = -(-1) - 3 = 1 - 3 = -2 \).
• When \( x = 5 \) (included, closed circle), \( f(5) = -5 - 3 = -8 \).

Plot the open circle at \((-1, -2)\) and closed circle at \((5, -8)\), then draw the line between them.

Answer:

(Graphing steps as above: first segment from open \((-4, 3)\) to closed \((-1, -6)\), second segment from open \((-1, -2)\) to closed \((5, -8)\))