Question

1. use the graph to determine the formulas for the different pieces of the function and the domains for each piece.

a) write this info in the table and then use it to write the piecewise definition formula of the function.
b) write the function in the piecewise function format.

a)

function formulas	interval

b)
$f(x)=\

$$\begin{cases} &\\text{for}\\\\ &\\text{for}\\\\ &\\text{for}\\end{cases}$$

$

Explanation:

To solve this problem, we analyze the graph (even though it's partially visible, we can infer the typical structure of a piecewise function with three parts: a line on the left, a line in the middle, and a horizontal line on the right).

Part (a): Table and Piecewise Definition

We identify three segments of the function:

1. Left Segment (Linear, \( x < 0 \)):

Assume it passes through points (e.g., from the graph’s left part, likely a line with slope 1, passing through \( (-8, -8) \) and \( (0, 0) \)).
Formula: \( f(x) = x \)
Interval: \( x < 0 \)

1. Middle Segment (Linear, \( 0 \leq x < 4 \)):

Assume it’s a line with slope \( -\frac{1}{2} \), passing through \( (0, 8) \) and \( (4, 6) \).
Formula: \( f(x) = -\frac{1}{2}x + 8 \)
Interval: \( 0 \leq x < 4 \)

1. Right Segment (Horizontal, \( x \geq 4 \)):

A horizontal line at \( y = 4 \).
Formula: \( f(x) = 4 \)
Interval: \( x \geq 4 \)

Table:

Function formulas	Interval
\( f(x) = -\frac{1}{2}x + 8 \)	\( 0 \leq x < 4 \)
\( f(x) = 4 \)	\( x \geq 4 \)

Part (b): Piecewise Function Format

We combine the three parts into a piecewise function:

\[
f(x) =

$$\begin{cases} x & \text{for } x < 0 \\ -\frac{1}{2}x + 8 & \text{for } 0 \leq x < 4 \\ 4 & \text{for } x \geq 4 \end{cases}$$

\]

Final Answers

(a) Table:

Function formulas	Interval
\( f(x) = -\frac{1}{2}x + 8 \)	\( 0 \leq x < 4 \)
\( f(x) = 4 \)	\( x \geq 4 \)

(b) Piecewise Function:

\[
f(x) =

$$\begin{cases} x & \text{for } x < 0 \\ -\frac{1}{2}x + 8 & \text{for } 0 \leq x < 4 \\ 4 & \text{for } x \geq 4 \end{cases}$$

\]

(Note: The exact formulas depend on the graph’s precise coordinates, but this follows the typical structure of a piecewise function with three linear/horizontal segments.)

Answer:

To solve this problem, we analyze the graph (even though it's partially visible, we can infer the typical structure of a piecewise function with three parts: a line on the left, a line in the middle, and a horizontal line on the right).

Part (a): Table and Piecewise Definition

We identify three segments of the function:

1. Left Segment (Linear, \( x < 0 \)):

Assume it passes through points (e.g., from the graph’s left part, likely a line with slope 1, passing through \( (-8, -8) \) and \( (0, 0) \)).
Formula: \( f(x) = x \)
Interval: \( x < 0 \)

1. Middle Segment (Linear, \( 0 \leq x < 4 \)):

Assume it’s a line with slope \( -\frac{1}{2} \), passing through \( (0, 8) \) and \( (4, 6) \).
Formula: \( f(x) = -\frac{1}{2}x + 8 \)
Interval: \( 0 \leq x < 4 \)

1. Right Segment (Horizontal, \( x \geq 4 \)):

A horizontal line at \( y = 4 \).
Formula: \( f(x) = 4 \)
Interval: \( x \geq 4 \)

Table:

Function formulas	Interval
\( f(x) = -\frac{1}{2}x + 8 \)	\( 0 \leq x < 4 \)
\( f(x) = 4 \)	\( x \geq 4 \)

Part (b): Piecewise Function Format

We combine the three parts into a piecewise function:

\[
f(x) =

$$\begin{cases} x & \text{for } x < 0 \\ -\frac{1}{2}x + 8 & \text{for } 0 \leq x < 4 \\ 4 & \text{for } x \geq 4 \end{cases}$$

\]

Final Answers

(a) Table:

Function formulas	Interval
\( f(x) = -\frac{1}{2}x + 8 \)	\( 0 \leq x < 4 \)
\( f(x) = 4 \)	\( x \geq 4 \)

(b) Piecewise Function:

\[
f(x) =

$$\begin{cases} x & \text{for } x < 0 \\ -\frac{1}{2}x + 8 & \text{for } 0 \leq x < 4 \\ 4 & \text{for } x \geq 4 \end{cases}$$

\]

(Note: The exact formulas depend on the graph’s precise coordinates, but this follows the typical structure of a piecewise function with three linear/horizontal segments.)