Question

a dog groomer charges her clients by the weight of the dog. the following peicewise function represents the total charge, f(x), for a dog weighing x pounds

Answer 1

It seems like you're looking to describe a piecewise function for the charges that a dog groomer applies based on the weight of the dog. Here's an example of how such a function might be structured:

Let's say the groomer has the following pricing:

- For dogs weighing 0 to 10 pounds: \( f(x) = 20 \) (a flat fee)
- For dogs weighing 11 to 30 pounds: \( f(x) = 20 + 1.50(x - 10) \) (base fee plus an additional $1.50 per pound over 10)
- For dogs weighing 31 to 50 pounds: \( f(x) = 20 + 30 + 1.00(x - 30) \) (if the dog weighs more than 30 pounds, it includes the previous charges plus $1.00 per pound over 30)
- For dogs over 50 pounds: \( f(x) = 50 + 0.80(x - 50) \) (if the dog exceeds 50 pounds, it adds $0.80 for each pound over 50)

Now, we can define the piecewise function more formally:

\[
f(x) =
\begin{cases}
20 & \text{if } 0 < x \leq 10 \\
20 + 1.50(x - 10) & \text{if } 10 < x \leq 30 \\
50 + 1.00(x - 30) & \text{if } 30 < x \leq 50 \\
80 + 0.80(x - 50) & \text{if } x > 50
\end{cases}
\]

Make sure to adjust the specific numbers and slopes according to the actual pricing of your dog grooming. Let me know if you'd like further assistance with this!