Question

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

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

$
line closed circle open circle
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 \( f(x) = x - 7 \) for \( -1 < x < 2 \)

• Find the endpoints:
• When \( x = -1 \) (not included, open circle), \( f(-1) = -1 - 7 = -8 \).
• When \( x = 2 \) (not included, open circle), \( f(2) = 2 - 7 = -5 \).
• This is a linear function with slope \( 1 \), so we draw a line segment from \( (-1, -8) \) (open circle) to \( (2, -5) \) (open circle).

Step2: Analyze the second piece \( f(x) = x - 6 \) for \( 2 < x \leq 5 \)

• Find the endpoints:
• When \( x = 2 \) (not included, open circle), \( f(2) = 2 - 6 = -4 \).
• When \( x = 5 \) (included, closed circle), \( f(5) = 5 - 6 = -1 \).
• This is a linear function with slope \( 1 \), so we draw a line segment from \( (2, -4) \) (open circle) to \( (5, -1) \) (closed circle).

Answer:

To graph the piecewise function:

1. For \( -1 < x < 2 \), draw a line segment (using the "Line" tool) with open circles at \( (-1, -8) \) and \( (2, -5) \).
2. For \( 2 < x \leq 5 \), draw a line segment (using the "Line" tool) with an open circle at \( (2, -4) \) and a closed circle at \( (5, -1) \).