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The problem in the image involves creating a piecewise function to describe the postal charges for a letter weighing $x$ ounces.

### Problem Breakdown:
- The postal service charges $0.37 for a letter weighing up to 1 ounce.
- For each additional ounce up to 5 ounces, an extra charge of $0.26 is added.
- The piecewise function will handle these different cases.

### Let's construct the piecewise function step by step:

#### 1. First case ( $ 0 < x \leq 1 $ ):
- If the weight is less than or equal to 1 ounce, the charge is $0.37.

#### 2. Second case ( $ 1 < x \leq 2 $ ):
- For a letter weighing more than 1 ounce but less than or equal to 2 ounces, the charge is the base rate ($0.37) plus one additional $0.26, which equals $0.37 + 0.26 = 0.63$.

#### 3. Third case ( $ 2 < x \leq 3 $ ):
- For a letter weighing more than 2 ounces but less than or equal to 3 ounces, the charge is $0.37 + 2 \times 0.26 = 0.37 + 0.52 = 0.89$.

#### 4. Fourth case ( $ 3 < x \leq 4 $ ):
- For a letter weighing more than 3 ounces but less than or equal to 4 ounces, the charge is $0.37 + 3 \times 0.26 = 0.37 + 0.78 = 1.15$.

#### 5. Fifth case ( $ 4 < x \leq 5 $ ):
- For a letter weighing more than 4 ounces but less than or equal to 5 ounces, the charge is $0.37 + 4 \times 0.26 = 0.37 + 1.04 = 1.41$.

### Piecewise Function:

\[
C(x) = 
\begin{cases} 
0.37 & \text{if } 0 < x \leq 1 \\
0.63 & \text{if } 1 < x \leq 2 \\
0.89 & \text{if } 2 < x \leq 3 \\
1.15 & \text{if } 3 < x \leq 4 \\
1.41 & \text{if } 4 < x \leq 5
\end{cases}
\]

This function accurately describes the postal charge for a letter weighing between 0 and 5 ounces.

Would you like more details or have any other questions?

Here are 5 related questions you might find helpful:
1. How do we compute the cost for weights beyond 5 ounces?
2. What if the postal rate increases by a fixed percentage annually?
3. How would this function change if the base charge was different?
4. Can you model the total cost for mailing multiple letters?
5. How do piecewise functions like this apply to real-world economics?

**Tip**: Piecewise functions are great for modeling situations with different conditions, like tiered pricing or tax brackets.

The postal service of a country charges $0.37 postage for a letter weighing up to 1 ounce, plus $0.26 for each additional ounce up to 5 ounces. Letters weighing more than 5 ounces do not qualify for these rates. Write a piecewise-defined function C for the postage charged for a letter weighing x ounces.

This problem involves constructing a piecewise function to represent postal charges for letters weighing between 1 and 5 ounces. The base charge is $0.37 for the first ounce, with an additional $0.26 per ounce. Learn how to model this using a piecewise function.

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