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Let's break down and solve the problems from the worksheet you uploaded. The topic is **Piecewise-Defined Functions**, and here’s how each question can be approached:

### 1. **Piecewise Function for Phone Company Plan**
- A phone company offers a monthly plan for $10. The plan includes 2 megabytes (MB) of data and charges $0.10 per MB for additional data usage beyond 2 MB.
- The piecewise-defined function $ M(x) $, where $ x $ represents the megabytes used, can be written as:
  \[
  M(x) = 
  \begin{cases} 
  10 & \text{if } 0 \leq x \leq 2 \\
  10 + 0.10(x - 2) & \text{if } x > 2
  \end{cases}
  \]
  This means the user pays a flat $10 for up to 2 MB, and for each MB beyond 2, an additional $0.10 is added.

### 2. **Graphing Piecewise Function and Analyzing**
The given function is:
\[
f(x) = 
\begin{cases} 
2 & \text{if } -4 \leq x \leq -2 \\
x + 2 & \text{if } -2 < x < 3 \\
-3x + 12 & \text{if } 3 \leq x \leq 5 
\end{cases}
\]
- **Domain and Range**: 
  - Domain: $ -4 \leq x \leq 5 $
  - Range: $ -3 \leq y \leq 12 $
- **Intervals**:
  - From $ -4 \leq x \leq -2 $, the function is constant (flat line) at $ y = 2 $.
  - From $ -2 < x < 3 $, the function is increasing because the slope of $ x + 2 $ is positive.
  - From $ 3 \leq x \leq 5 $, the function is decreasing due to the negative slope of $ -3x + 12 $.

### 3. **Rule for Graph’s Piecewise-Defined Function**
- The graph shows three distinct linear segments:
  \[
  g(x) = 
  \begin{cases} 
  0.5x + 3 & \text{if } -6 \leq x < -2 \\
  2 & \text{if } -2 \leq x \leq 1 \\
  x - 1 & \text{if } 1 < x \leq 6
  \end{cases}
  \]

### 4. **Piecewise Function for $ f(x) = |2x - 8| $**
- To write this absolute value function as a piecewise function, we need to split it into two cases based on when $ 2x - 8 $ is non-negative or negative:
  \[
  f(x) = 
  \begin{cases} 
  2x - 8 & \text{if } x \geq 4 \\
  -(2x - 8) = -2x + 8 & \text{if } x < 4
  \end{cases}
  \]

### 5. **Shipping Cost Based on Weight**
- The function describes the cost of shipping based on weight $ x $, with a base rate of $3 for up to 5 pounds and $2 for each additional pound. This can be written as:
  \[
  P(x) = 
  \begin{cases} 
  3 & \text{if } 0 < x \leq 5 \\
  3 + 2(x - 5) & \text{if } x > 5
  \end{cases}
  \]
  **Domain**: $ 0 < x \leq 11 $ (since the maximum given weight is 11 pounds)
  
  **Range**: $ 3 \leq P(x) \leq 13 $ (since the cost starts at $3 and increases to $13 for 11 pounds)

### 6. **Car Rental Model**
- The piecewise function for the car rental model, based on the conditions:
  \[
  R(x) = 
  \begin{cases} 
  35x & \text{if } x \leq 3 \\
  35x + 5(x - 3) & \text{if } x > 3
  \end{cases}
  \]
  This reflects $35 per day, with an additional $5 insurance fee per day after the third day.

Would you like to go into more detail on any of these questions, or need further clarification?

### Related Questions:
1. How do you determine if a function is increasing or decreasing in a piecewise function?
2. What steps are involved in converting absolute value functions into piecewise functions?
3. How do you find the domain and range of a piecewise function?
4. How can piecewise functions model real-world scenarios, like pricing plans?
5. How can you check the continuity of a piecewise-defined function?

### Tip:
Always double-check the boundary points in piecewise functions to ensure the conditions correctly match the function's behavior at those points.

A set of problems focused on piecewise-defined functions, including scenarios involving data plans, graphs of piecewise functions, absolute value functions, shipping cost calculations, and car rental pricing models.

This page covers a variety of problems involving piecewise-defined functions. Topics include writing piecewise functions for data plans, graphing piecewise functions, solving absolute value equations, calculating shipping costs based on weight, and car rental cost models. Perfect for students in Grades 9-11 studying Algebra II.

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