Question

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

$$\begin{cases}x - 4 & \\text{for } -2 < x < 1 \\\\ 3x - 7 & \\text{for } 1 < x < 4 \\end{cases}$$

$
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.
(graph axes with y and x axes, grid)

Explanation:

Step1: Analyze the first piece \( f(x)=x - 4 \) for \( -2 < x < 1 \)

• Find the endpoints:
• When \( x=-2 \) (not included, open circle), \( f(-2)=-2 - 4=-6 \).
• When \( x = 1 \) (not included, open circle), \( f(1)=1 - 4=-3 \).
• This is a linear function with slope \( 1 \), so we can plot the line segment between \( (-2, -6) \) (open circle) and \( (1, -3) \) (open circle) by connecting these two points with a line (since it's linear).

Step2: Analyze the second piece \( f(x)=3x - 7 \) for \( 1 < x < 4 \)

• Find the endpoints:
• When \( x = 1 \) (not included, open circle), \( f(1)=3(1)-7=-4 \).
• When \( x = 4 \) (not included, open circle), \( f(4)=3(4)-7 = 12 - 7 = 5 \).
• This is a linear function with slope \( 3 \), so we can plot the line segment between \( (1, -4) \) (open circle) and \( (4, 5) \) (open circle) by connecting these two points with a line (since it's linear).

Graphing Steps (Summary):

1. For \( y=x - 4 \), \( -2 < x < 1 \):

• Plot an open circle at \( (-2, -6) \) and \( (1, -3) \), then draw a line connecting these two open - circled points.

1. For \( y = 3x - 7 \), \( 1 < x < 4 \):

• Plot an open circle at \( (1, -4) \) and \( (4, 5) \), then draw a line connecting these two open - circled points.

(Note: Since this is a graphing problem, the final answer is the visual graph constructed as per the above steps. If we were to describe the key points:

• For \( f(x)=x - 4, -2 < x < 1 \): passes through points like when \( x = 0 \), \( f(0)=0 - 4=-4 \), so \( (0, -4) \) is on this segment (between \( (-2, -6) \) and \( (1, -3) \)).
• For \( f(x)=3x - 7, 1 < x < 4 \): when \( x = 2 \), \( f(2)=3(2)-7=-1 \); when \( x = 3 \), \( f(3)=3(3)-7 = 2 \), so \( (2, -1) \) and \( (3, 2) \) are on this segment (between \( (1, -4) \) and \( (4, 5) \)).)

Answer:

The graph consists of two line segments:

• One segment for \( y=x - 4 \) with open circles at \( (-2, -6) \) and \( (1, -3) \), passing through points like \( (0, -4) \).
• One segment for \( y = 3x - 7 \) with open circles at \( (1, -4) \) and \( (4, 5) \), passing through points like \( (2, -1) \) and \( (3, 2) \).