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Let's break down and solve the questions step by step:

### 1. (e) **Find a piecewise formula for \( f(x) \):** 
From the question, the graph is not visible, but in general, to find a piecewise function for \( f(x) \), you need to analyze different intervals of \( x \) based on the behavior of the graph (e.g., increasing, decreasing, or constant). Once the behavior changes in a specific interval, you write a different function for that interval. Without the graph or detailed information, it’s difficult to provide a specific answer here. If you have more information on the graph of \( f(x) \), we can proceed to create a piecewise function.

### 2. The graph of \( W(x) \) represents the depth of water in a reservoir as a function of days, \( x \), where \( 0 \leq x \leq 365 \).

#### (a) **What is the domain of this function? What is the range?** 
- **Domain:** 
  Since the function represents the depth of water for every day of the year, \( x \) is the number of days since the beginning of the year. Therefore, the domain is:
  \[
  0 \leq x \leq 365
  \]
  
- **Range:**
  From the graph, the lowest value of \( W(x) \) seems to be around 0 and the highest around 100. So, the range is approximately:
  \[
  0 \leq W(x) \leq 100
  \]

#### (b) **Find \( W(100) \) and \( W(200) \). What do these values represent?** 
- From the graph:
  - \( W(100) \approx 75 \)
  - \( W(200) \approx 25 \)
  
- These values represent the depth of water in the reservoir on day 100 and day 200, respectively. \( W(100) = 75 \) means that the water depth on day 100 is about 75 feet, and \( W(200) = 25 \) means that the water depth on day 200 is about 25 feet.

#### (c) **Find the value(s) of \( x \) such that \( W(x) = 50 \).**
From the graph, it appears that \( W(x) = 50 \) occurs at two points: once between day 0 and day 100 and once between day 200 and day 300. Estimating from the graph:
  - \( W(x) = 50 \) at approximately \( x = 50 \) and \( x = 275 \).

#### (d) **Determine the intervals on which \( W(x) \) is increasing and decreasing.**
- From the graph:
  - \( W(x) \) is increasing approximately from \( x = 0 \) to \( x = 75 \).
  - \( W(x) \) is decreasing from \( x = 75 \) to \( x = 150 \).
  - \( W(x) \) is increasing from \( x = 150 \) to \( x = 225 \).
  - \( W(x) \) is decreasing from \( x = 225 \) to \( x = 365 \).

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Let me know if you'd like any additional details or explanations!

Here are 5 related questions you could explore:
1. How can you find the maximum and minimum values of \( W(x) \) from the graph?
2. How do you calculate the average rate of change of \( W(x) \) over specific intervals?
3. Can you determine when \( W(x) \) is concave up or concave down?
4. What is the significance of the intercepts of \( W(x) \)?
5. How does the period of oscillation affect the behavior of \( W(x) \)?

**Tip:** In piecewise functions, always pay close attention to whether endpoints are included or excluded in each interval!

The graph of W(x) represents the depth of water in a reservoir, with questions on domain, range, piecewise function, and intervals of increase and decrease.

Learn how to interpret the graph of a function representing the depth of water in a reservoir over time. This problem includes finding domain, range, and intervals of increase and decrease, as well as constructing a piecewise function. Suitable for students in Grades 10-12.

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