Question

f(x) = \

$$\begin{cases} -x - 2 & \\text{for} & -6 < x < 2 \\\\ x - 7 & \\text{for} & 2 < x \\leq 6 \\end{cases}$$

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

Explanation:

Step1: Analyze the first piece: \( y = -x - 2 \), domain \( -6 < x < 2 \)

• Find two points. When \( x = -6 \) (not included, open circle), \( y = -(-6) - 2 = 4 \). When \( x = 2 \) (not included, open circle), \( y = -2 - 2 = -4 \).

Step2: Analyze the second piece: \( y = x - 7 \), domain \( 2 < x \leq 6 \)

• Find two points. When \( x = 2 \) (not included, open circle), \( y = 2 - 7 = -5 \). When \( x = 6 \) (included, closed circle), \( y = 6 - 7 = -1 \).

Step3: Plot the lines

• For \( y = -x - 2 \), draw a line between \( (-6, 4) \) (open) and \( (2, -4) \) (open).
• For \( y = x - 7 \), draw a line between \( (2, -5) \) (open) and \( (6, -1) \) (closed).

(Note: Since the problem is about graphing, the key is to identify the endpoints for each piecewise function and plot them with correct open/closed circles based on the domain inequalities.)

Answer:

To graph \( f(x) \):

1. For \( -6 < x < 2 \), graph \( y = -x - 2 \) (open circles at \( (-6, 4) \) and \( (2, -4) \), line connecting them).
2. For \( 2 < x \leq 6 \), graph \( y = x - 7 \) (open circle at \( (2, -5) \), closed circle at \( (6, -1) \), line connecting them).