Question

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

$$\begin{cases} x + 6 & \\text{for} & -4 < x < -1 \\\\ -2x + 7 & \\text{for} & -1 < x \\leq 5 \\end{cases}$$

$
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 \( y = x + 6 \) (domain \( -4 < x < -1 \))

Find two points on this line.

• When \( x = -4 \) (not included, open circle), \( y = -4 + 6 = 2 \).
• When \( x = -1 \) (not included, open circle), \( y = -1 + 6 = 5 \).

Plot these two open - circle points \((-4,2)\) and \((-1,5)\) and draw a line segment between them.

Step2: Analyze the second piece \( y=-2x + 7\) (domain \( -1 < x\leq5 \))

Find two points on this line.

• When \( x=-1\) (not included, open circle), \( y=-2\times(-1)+7 = 2 + 7=9\).
• When \( x = 5\) (included, closed circle), \( y=-2\times5 + 7=-10 + 7=-3\).

Plot the open - circle point \((-1,9)\) and the closed - circle point \((5, - 3)\) and draw a line segment between them.

Answer:

To graph the piece - wise function:

1. For \(y=x + 6\) (\(-4 < x < -1\)):

• Plot open circles at \((-4,2)\) and \((-1,5)\) and draw a line segment connecting them.

1. For \(y=-2x + 7\) (\(-1 < x\leq5\)):

• Plot an open circle at \((-1,9)\) and a closed circle at \((5,-3)\) and draw a line segment connecting them.