To join a dance lesson, you paid a one-time membership fee of $5 000, then $500 for every session. If they gave you one free session after every ten sessions and you only have attended a number of paid sessions that is a multiple of 10, determine the function that represents the average cost per session and identify if the function is a rational function.

Let $ x $ represent the number of paid sessions attended, where $ x $ is a multiple of 10. Here's how we will derive the function for the average cost per session:

### Total Cost:
- The one-time membership fee is $5,000.
- For every paid session, you pay $500.
- After every 10 paid sessions, you receive 1 free session. Thus, if you have attended $ x $ paid sessions, you have attended $ \frac{x}{10} $ free sessions.
- Therefore, the total number of sessions (including free ones) is $ x + \frac{x}{10} $.

The total cost for $ x $ paid sessions is:
\[
\text{Total cost} = 5000 + 500x
\]

### Average Cost per Session:
To find the average cost per session, divide the total cost by the total number of sessions (including free ones):
\[
\text{Average cost per session} = \frac{5000 + 500x}{x + \frac{x}{10}} = \frac{5000 + 500x}{1.1x}
\]

### Simplifying the Expression:
\[
\text{Average cost per session} = \frac{500(10 + x)}{1.1x} = \frac{500(10 + x)}{1.1x}
\]

### Final Function:
Thus, the function representing the average cost per session is:
\[
f(x) = \frac{500(10 + x)}{1.1x}
\]

### Is it a Rational Function?
Yes, this is a rational function because it is the ratio of two polynomials: the numerator $ 500(10 + x) $ is a linear polynomial, and the denominator $ 1.1x $ is also a linear polynomial.

Would you like more details on any of the steps or a graph of this function?

Here are 5 follow-up questions to expand on this:
1. How does the average cost per session change as the number of sessions increases?
2. What is the average cost per session after 20 paid sessions?
3. What happens to the average cost per session as $ x $ approaches infinity?
4. How does the presence of free sessions affect the average cost?
5. How could the model change if the membership fee were lower?

**Tip**: Always simplify expressions step by step to avoid errors, especially when working with rational functions!

To join a dance lesson, you paid a one-time membership fee of $5,000, then $500 for every session. If they gave you one free session after every ten sessions and you only have attended a number of paid sessions that is a multiple of 10, determine the function that represents the average cost per session and identify if the function is a rational function.

This solution explains how to calculate the average cost per session for a dance membership with a one-time fee and per-session charges. It includes a rational function that accounts for free sessions given after every ten paid sessions. Suitable for high school students in Grades 10-12.

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