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The problem asks to express the postage cost \( P(x) \) as a piecewise function of the weight \( x \) (in ounces) of a first-class letter weighing up to 3.5 ounces. The postage rate for 2014 is 49 cents for the first ounce (or less) and 21 cents for each additional ounce (or part thereof). Here's how we can break it down:

- For \( 0 < x \leq 1 \), the cost is just the base rate, 49 cents.
- For \( 1 < x \leq 2 \), you add 21 cents to the base rate (49 + 21 = 70 cents).
- For \( 2 < x \leq 3 \), you add another 21 cents to the previous rate (70 + 21 = 91 cents).
- For \( 3 < x \leq 3.5 \), you add one more 21 cents to the previous rate (91 + 21 = 112 cents).

Now, expressing this as a piecewise function \( P(x) \) in dollars:

\[
P(x) =
\begin{cases}
0.49 & \text{if } 0 < x \leq 1 \\
0.70 & \text{if } 1 < x \leq 2 \\
0.91 & \text{if } 2 < x \leq 3 \\
1.12 & \text{if } 3 < x \leq 3.5 \\
\end{cases}
\]

This function breaks down the postage cost based on the weight \( x \) of the letter.

Would you like more details or have any other questions?

Here are 5 related questions for deeper understanding:
1. How would the piecewise function change if the rates for additional ounces increased to 25 cents per ounce?
2. What would the total cost be for a letter weighing exactly 3 ounces?
3. How does the concept of a piecewise function help in modeling real-world pricing scenarios like this one?
4. Can you find a different real-life scenario where a piecewise function might apply, similar to postage rates?
5. How would you graph this piecewise function to visualize the cost against the weight?

**Tip:** Piecewise functions are ideal for modeling situations with different rules over different intervals. Use them to express tiered pricing, like shipping or utility rates.

The 2014 domestic postage rate for first-class letters weighing 3.5 oz or less is 49 cents for the first ounce (or less), plus 21 cents for each additional ounce (or part of an ounce). Express the postage P as a piecewise defined function of the weight x of a letter, with 0 < x ≤ 3.5. (Express your answer in dollars.)

Explore how the 2014 postage rates for first-class letters are structured as a piecewise function depending on the weight of the letter. Rates start at 49 cents for up to one ounce and increase by 21 cents for each additional ounce up to 3.5 ounces. This problem provides a practical application of piecewise and linear functions.

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