A certain paper book sells for $25. The author receives a royalty of 12% on the first 5000 copies sold

Question

A certain paper book sells for $25. The author receives a royalty of 12% on the first 5000 copies sold, 15% on the next 10000 copies sold, and 20% for any additional copies sold. Write R, the amount of royalties in dollars as a function of x, the number of copies sold.

This is what I have so far.

$$R(x) = \begin{cases}25*(0.12)(x)& 0 \le x \le 5000\\ 25*(0.12)(5000) + 25*(0.15)(x-5000) & 5000 < x \le 15000 \\ 25*(0.12)(5000) + 25*(0.15)(15000) + 25*(0.2)(x-15000)& x > 15000 \end{cases} $$

I'm not sure if they want it all in one function or if that is even possible, anyways any help would be appreciated.

Answer 1

This is a fine function. It gives a single output for every input. The $15000$ in the last line should be $10000$ because that is the number of copies the $15\%$ royalty applies to. They may want you to do the arithmetic and make the last line look like $5x-k$ for some $k$ but your version makes it clear where the terms come from.

Answer 2

Yes, your answer is fine. Two notes:

$1)$ The function is called a piece-wise function.

$2)$ Since Revenue = Price x Quantity and/or Royalty = Price x Quantity, it would be better to express:

$$R(x) = \begin{cases} (25\cdot0.12)(x), & 0 \le x \le 5000\\ (25\cdot0.12)(5000) + (25\cdot0.15)(x-5000), & 5000 < x \le 15000 \\ (25\cdot0.12)(5000) + (25\cdot0.15)(15000) + (25\cdot0.2)(x-15000), & x > 15000 \end{cases}$$