https://file.aiquickdraw.com/mathbot/file/17277373201399cveclo7.jpg

The page presents two graphs, and the task asks for the following for each graph:

### Steps for Each Graph:
1. **Identify the input and output** and define a variable to represent each.
2. **Determine the practical domain** and **practical range** using proper notation.

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### **Graph 1: Detroit Population Over Time**
- **Input/Output and Variables:**
  - **Input**: Time, represented in years up to 1990 (denoted as $ t $).
  - **Output**: Population of Detroit, measured in thousands (denoted as $ P(t) $).
  
- **Practical Domain**:
  - The graph covers time from $ t = 0 $ (starting point, unspecified year) to $ t = 10 $ (10 years).
  - Domain: $ [0, 10] $
  
- **Practical Range**:
  - Population starts at about 1,600,000 and decreases to 800,000.
  - Range: $ [800,000, 1,600,000] $

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### **Graph 2: Amount of Money Over Weeks**
- **Input/Output and Variables:**
  - **Input**: Number of weeks, denoted as $ n $.
  - **Output**: Amount of money in dollars, denoted as $ A(n) $.
  
- **Practical Domain**:
  - Number of weeks $ n $ goes from 0 to 12.
  - Domain: $ [0, 12] $
  
- **Practical Range**:
  - The amount of money ranges from $0 to $275.
  - Range: $ [0, 275] $

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Would you like more detailed explanations for each? 

Here are five questions for further exploration:
1. How does the practical domain differ from the mathematical domain in function analysis?
2. Can the population trend of Detroit be modeled with a specific type of function?
3. How would you interpret the rate of population decrease from the graph?
4. What might the slope represent in the second graph about money accumulation?
5. Could the money graph be used to model savings growth, and what assumptions would be needed?

**Tip**: Always consider the context of a graph when determining its practical domain and range; they reflect the real-world limits of the situation.

For each of the given graphs and table: a. Identify the input and output and define a variable to represent each. b. Determine the practical domain and practical range for each graph. Use proper notation.

Analyze two graphs: Detroit population over time and money over weeks. Identify the input, output, variables, practical domain, and practical range for each. Suitable for high school students studying functions.

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