Jill has a business organizing children’s birthday parties. For a party with up to six children, she charges $\$50$. She charges another $\$5$ for each additional guest.

1. Write a piecewise-defined function $M$ to represent the money Jill receives for organizing a birthday party for $x$ children. Be sure to include any restrictions on the domain. Check your function using several different numbers of children.

   Hint (a):

   For $1$ to $6$ children the cost is a constant function.

   More Help (a):

   Use this pattern to write a rule for seven or more children.
   $7$ children: $50 + 5 · 1$
   $8$ children: $50 + 5 · 2$
   $9$ children: $50 + 5 · 3$

   Answer (a):

   $M(x)=\begin{cases}50\:\:\:\text{if }1\leq x \leq 6\\50+5(x-6)\:\:\:\text{if }x\geq 7 \end{cases}$

2. The party favors and treats cost $\$3.00$ per child. Write a function $E$ to represent the expenses for a party for $x$ children.

   Answer (b):

   $E(x) = 3x$

3. What is Jill’s net income, $N$, from a party for $x$ children in terms of $M$ and $E$? That is, how much money will she have after she pays for the party favors and treats?

   Hint (c):

   Write an expression that subtracts her expenses from her income.

4. Write a piecewise-defined function for $N$ including restrictions on the domain, and check it by testing values for different numbers of children.

   Hint (d):

   Combine the answers from parts (a) − (c) into a piecewise function.