Question

which graph models the piecewise functions shown below:
$f(x) = \

$$\begin{cases} -\\frac{1}{3}x - 3 & -9 \\leq x \\leq -3 \\\\ -1 & -3 < x \\leq 1 \\\\ 3x - 3 & 1 < x < 4 \\end{cases}$$

$

Explanation:

Step1: Analyze the first piece

For \( f(x)=-\frac{1}{3}x - 3 \) with \( -9\leq x\leq - 3 \).

• When \( x=-9 \), \( f(-9)=-\frac{1}{3}(-9)-3 = 3 - 3=0 \).
• When \( x = - 3 \), \( f(-3)=-\frac{1}{3}(-3)-3=1 - 3=-2 \).

This is a line segment with slope \( -\frac{1}{3} \), starting at \( (-9,0) \) and ending at \( (-3,-2) \), including both endpoints.

Step2: Analyze the second piece

For \( f(x)=-1 \) with \( -3\lt x\leq1 \).
This is a horizontal line segment at \( y = - 1 \), starting just to the right of \( x=-3 \) (open circle at \( x = - 3 \)) and ending at \( x = 1 \) (closed circle at \( x = 1 \)).

Step3: Analyze the third piece

For \( f(x)=3x - 3 \) with \( 1\lt x\lt4 \).

• When \( x = 1 \) (approaching from the right), \( f(1)=3(1)-3 = 0 \) (open circle at \( x = 1 \)).
• When \( x = 4 \) (approaching from the left), \( f(4)=3(4)-3=9 \) (open circle at \( x = 4 \)).

This is a line segment with slope \( 3 \), starting just to the right of \( x = 1 \) and ending just to the left of \( x = 4 \).

To identify the correct graph, we check the key points and the behavior of each piece:

• The first piece connects \( (-9,0) \) to \( (-3,-2) \).
• The second piece is horizontal at \( y=-1 \) from just right of \( x = - 3 \) to \( x = 1 \).
• The third piece has a positive slope, starting at an open circle at \( (1,0) \) (since at \( x = 1 \), the second piece ends with a closed circle at \( (1,-1) \) and the third piece starts with an open circle at \( (1,0) \)) and going up to near \( (4,9) \).

Answer:

To determine the correct graph, we check the key points and the behavior of each piece of the piecewise function. The correct graph should have:

1. A line segment with slope \(-\frac{1}{3}\) from \((-9, 0)\) to \((-3, -2)\) (including both endpoints).
2. A horizontal line segment at \(y = -1\) from just right of \(x=-3\) (open circle) to \(x = 1\) (closed circle).
3. A line segment with slope \(3\) from just right of \(x = 1\) (open circle at \((1,0)\)) to just left of \(x = 4\) (open circle at \((4,9)\)).

(Note: Since the actual graphs are not provided here, the above analysis helps in identifying the correct graph by matching these key features.)