https://file.aiquickdraw.com/mathbot/file/1725932009962q7n8thc9.jpg

Let's break down the problem and the piecewise function based on the given information.

The problem states:

1. **Cost for fewer than 75 people:**
   - The cost is \$25.00 per person.
   - There is a service charge of \$133.75.
   
   So, the total cost function for fewer than 75 people (i.e., $ n < 75 $) is:
   \[
   T(n) = 25n + 133.75
   \]

2. **Cost for 75 or more people:**
   - The cost per person drops to \$21.75.
   - The service charge is reduced to \$66.00.

   So, the total cost function for 75 or more people (i.e., $ n \geq 75 $) is:
   \[
   T(n) = 21.75n + 66.00
   \]

### Piecewise-defined linear function:
We can now write the piecewise function as:

\[
T(n) = 
\begin{cases} 
25n + 133.75 & \text{if } n < 75 \\
21.75n + 66.00 & \text{if } n \geq 75
\end{cases}
\]

Would you like more details or clarifications on any part of the solution?

Here are some related questions to further explore:

1. How does the service charge impact the total cost in each case?
2. What happens to the overall cost if the number of people increases from 74 to 75?
3. Can we graph this piecewise function to visualize the cost differences?
4. How would the function change if the service charges were the same in both cases?
5. What is the total cost for exactly 100 people?

**Tip:** When working with piecewise functions, always check the boundary points (like $ n = 75 $) to ensure the correct formula is applied.

A restaurant offers a catering service which costs $25.00 per person with a $133.75 service charge. For parties of 75 or more people, a group discount applies, and the cost is $21.75 per person along with the service charge dropping to $66.00. Write a piecewise-defined linear function which calculates the total cost T of the catering service which serves n people.

Learn how to write a piecewise-defined linear function for catering service costs, considering different pricing for less than 75 people and 75 or more people. Understand key algebraic concepts with formulas for calculating the total cost.

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