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 $x = 365$.

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!