3. A videoke machine can be rented for P1,000 for three days, but for the fourth day onwards, an additional cost of P400 per day is added. Represent the cost of renting a videoke machine as a piecewise function of the number of days it is rented and plot its graph. Answer:

Here's the graph of the piecewise function \( C(d) \) representing the cost of renting a videoke machine:

- For the first 3 days, the cost is a constant P1,000, represented by the horizontal line.
- From the 4th day onwards, the cost increases by P400 for each additional day, which is reflected by the upward-sloping line starting at day 4.

Would you like further details or have any questions? Here are some related questions to explore:

1. How does the cost function change if the initial 3-day rental cost is increased to P1,500?
2. How would the graph look if the additional daily cost after the third day was P500 instead of P400?
3. What is the total cost if the videoke machine is rented for 7 days?
4. How would the function change if there was a discount after renting for more than 7 days?
5. Can you represent the cost function using a different base rate for the first few days?
6. How does the piecewise function model compare to real-world rental pricing scenarios?
7. What happens if you introduce a maximum rental period of 10 days, with a fixed cost afterward?
8. How would the cost function change if a penalty fee was added for rentals over 10 days?

**Tip:** Understanding how to graph piecewise functions is essential for visualizing how costs or other quantities change over different intervals.

Explore the cost structure of renting a videoke machine using a piecewise function. Understand how the cost varies for different rental periods, with a fixed cost for the first 3 days and an additional daily charge thereafter. Detailed graph and analysis provided.

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