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Part 1
The postal service of a country charges $ 0.34 postage for a letter weighing up to 1​ ounce, plus $ 0.24 for each additional ounce up to 5 ounces. Letters weighing more than 5 ounces do not qualify for these rates.
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Part 1
​(a) Write a​ piecewise-defined function C for the postage charged for a letter weighing x ounces.
Upper C left parenthesis x right parenthesis equals left brace Start 5 By 2 Matrix 1st Row 1st Column $ nothing 2nd Column if 0 less than x less than or equals 1 2nd Row 1st Column $ nothing 2nd Column if 1 less than x less than or equals 2 3rd Row 1st Column $ nothing 2nd Column if 2 less than x less than or equals 3 4st Row 1st Column $ nothing 2nd Column if 3 less than x less than or equals 4 5st Row 1st Column $ nothing 2nd Column if 4 less than x less than or equals 5 EndMatrix

To define the postage charged for a letter weighing $ x $ ounces, we need to create a piecewise function that accounts for the different rates based on the weight of the letter. 

### Understanding the Problem:
- For a letter weighing up to 1 ounce, the cost is $0.34.
- For each additional ounce up to 5 ounces, there is an additional charge of $0.24 per ounce.
- Letters weighing more than 5 ounces do not qualify for these rates, but this part is not included in the function since the domain is restricted up to 5 ounces.

### Defining the Piecewise Function:

The cost function $ C(x) $ for the postage can be defined as:

\[
C(x) = 
\begin{cases} 
0.34 & \text{if } 0 < x \leq 1 \\
0.34 + 0.24 \times (x - 1) & \text{if } 1 < x \leq 2 \\
0.34 + 0.24 \times (x - 1) & \text{if } 2 < x \leq 3 \\
0.34 + 0.24 \times (x - 1) & \text{if } 3 < x \leq 4 \\
0.34 + 0.24 \times (x - 1) & \text{if } 4 < x \leq 5 \\
\end{cases}
\]

Simplifying further, we can rewrite $ C(x) $ to a more compact form:

\[
C(x) = 
\begin{cases} 
0.34 & \text{if } 0 < x \leq 1 \\
0.58 & \text{if } 1 < x \leq 2 \\
0.82 & \text{if } 2 < x \leq 3 \\
1.06 & \text{if } 3 < x \leq 4 \\
1.30 & \text{if } 4 < x \leq 5 \\
\end{cases}
\]

### Conclusion:

The piecewise function $ C(x) $ representing the cost of postage is:

\[
C(x) = 
\begin{cases} 
0.34 & \text{if } 0 < x \leq 1 \\
0.58 & \text{if } 1 < x \leq 2 \\
0.82 & \text{if } 2 < x \leq 3 \\
1.06 & \text{if } 3 < x \leq 4 \\
1.30 & \text{if } 4 < x \leq 5 \\
\end{cases}
\]

Would you like more details on how to compute each case, or do you have any questions?

### Further Questions for Understanding:
1. What is the cost of postage for a letter weighing exactly 1.5 ounces?
2. How does the postage cost change as the weight increases beyond 1 ounce?
3. What would the function look like for letters weighing more than 5 ounces?
4. Can you graph the piecewise function to visualize the cost progression?
5. How does this piecewise function relate to other step functions you've encountered?

### Tip:
When dealing with piecewise functions, always carefully consider the conditions and simplify the function expressions for clarity and ease of calculation.

The postal service of a country charges $0.34 postage for a letter weighing up to 1 ounce, plus $0.24 for each additional ounce up to 5 ounces. Letters weighing more than 5 ounces do not qualify for these rates. (a) Write a piecewise-defined function C for the postage charged for a letter weighing x ounces.

Learn how to write a piecewise-defined function for calculating postage based on weight. This problem covers key concepts in piecewise functions and linear equations. Suitable for students in Grades 9-12.

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