Question

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

$$\begin{cases} -x - 1 & \\text{for} & -4 < x < 1 \\\\ -2x + 1 & \\text{for} & 1 < x \\leq 5 \\end{cases}$$

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

Explanation:

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

• Find two points on this line.
• When $x = -4$ (not included, open circle), $f(-4) = -(-4) - 1 = 4 - 1 = 3$.
• When $x = 1$ (not included, open circle), $f(1) = -1 - 1 = -2$.
• So we have a line segment (open endpoints) between $(-4, 3)$ and $(1, -2)$ with slope $-1$.

Step2: Analyze the second piece $f(x) = -2x + 1$ for $1 < x \leq 5$

• Find two points on this line.
• When $x = 1$ (not included, open circle), $f(1) = -2(1) + 1 = -1$.
• When $x = 5$ (included, closed circle), $f(5) = -2(5) + 1 = -10 + 1 = -9$.
• So we have a line segment (open at $x = 1$, closed at $x = 5$) between $(1, -1)$ and $(5, -9)$ with slope $-2$.

Step3: Graph the pieces

• For the first piece, plot the open circles at $(-4, 3)$ and $(1, -2)$, then draw a line through them.
• For the second piece, plot the open circle at $(1, -1)$ and closed circle at $(5, -9)$, then draw a line through them.

(Note: Since this is a text - based response, the actual graphing would be done on the provided axes by following the above steps of plotting points and drawing line segments with the correct endpoint types (open/closed circles).)

Answer:

To graph the piece - wise function:

1. For $y=-x - 1, - 4\lt x\lt1$:

• It is a line with slope $-1$. The left - hand endpoint (not included) is at $x=-4,y = 3$ (open circle), and the right - hand endpoint (not included) is at $x = 1,y=-2$ (open circle). Draw a line segment between these two open - circle points.

1. For $y=-2x + 1,1\lt x\leq5$:

• It is a line with slope $-2$. The left - hand endpoint (not included) is at $x = 1,y=-1$ (open circle), and the right - hand endpoint (included) is at $x = 5,y=-9$ (closed circle). Draw a line segment between this open - circle and closed - circle point.