1. ## whats the function?

Three students in an outdoor pursuits program collaborated to write an adventure travel guidebook for Alberta and BC. Two local publishers are interested in marketing and publishing it and both anticipate retailing the book for $20. Publisher "A" offers a straight royalty of 10% (on the retail price) on all book sales. Publisher "B" offers an 8% royalty for any books sold up to 5,000 copies and will then increase the royalty to 15% for any copies sold in excess of 5,000. a) Describe the type of function that will be used to model the author income offered by each publisher. b) Find the functions A(x) and B(x). Notice that function A(x) is a piecewise-defined function. 2. Hello, Cockchestner007! Three students wrote an adventure travel guidebook for Alberta and BC. Two local publishers are interested in marketing it and both anticipate retailing the book for$20.

Publisher "A" offers a straight royalty of 10% (on the retail price) on all book sales.

Publisher "B" offers an 8% royalty for any books sold up to 5,000 copies
and will then increase the royalty to 15% for any copies sold in excess of 5,000.

a) Describe the type of function that will model the authors' income offered by each publisher.

Publisher A: The authors get $\displaystyle 10\% \times \$20 \:=\:\$2$ for each book sold.
. . . . . . . . . This is a linear function.

Publisher B: The authors get $\displaystyle 8\% \times \$20 \:=\:\$1.60$ for each book sold up to 5,000 books.
. . . . . . . . . They get $\displaystyle 15\% \times \$2 \:=\:\$3$ for each book in excess of 5,000.
. . . . . . . . . This is a lnear piecewise function.

b) Find the functions $\displaystyle A(x)$ and $\displaystyle B(x).$
. . Note that function $\displaystyle {\color{red}B(x)}$ is a piecewise-defined function.
Let $\displaystyle x$ = number of books sold.

Publisher A: .$\displaystyle A(x) \;=\;2x$

Publisher B: this is quite tricky.

For sales up to 5,000, they get: .$\displaystyle \$1.60$per book. . .$\displaystyle B(x) \;=\;1.6x\quad x \leq 5000$For sales in excess of 5,000, they get: .$\displaystyle \$3$ per book.
. . $\displaystyle B(x) \:=\:3(x-5000) \quad x > 5000$

If $\displaystyle x > 5000$, they still get $1.60 for the first 5000 books: 8,000 dollars, . . plus$3 each for all books over 5000: $\displaystyle 3(x-5000)$ dollars.

Hence, for $\displaystyle x > 5000,\;B(x) \:=\:8000 + 3(x-5000) \;=\;3x - 7000$ dollars.

Therefore: .$\displaystyle B(x) \;=\;\bigg\{\,\begin{array}{cccc} 2x && x \leq 5000 \\ 3x - 7000 && x > 5000 \end{array}$