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!