c Write an expression that subtracts her expenses from her income.
D
d Combine the answers from Parts A and B into a piecewise function.
A
a M(x)= 50, & -3pt {x|1,2,...,6} 50+(x-6), & -3pt {x|7,8,...}
B
b E(x)=3x, {x|1,2,3,...}
C
c N(x)=M(x)-E(x)
D
d N(x)= 50-3x, & -3pt {x|1,2,...,6} 2x+20, & -3pt {x|7,8,...}
Practice makes perfect
a From the exercise, we know that for 1 child up to and including 6 children, Jill charges $50. Let's graph this. Notice that our graph will only be defined for natural numbers of x Therefore, we get a discrete function.
We can write this function as m(x)= 50 with a domain of {x | 1,2,...,6}. After the sixth child, Jill charges an extra $5 per child which means we get the second part of our piecewise function. Like before, it is only defined for natural numbers.
This second function starts at 55 when x=7 and has a slope of 5. We can write this as the equation m(x)= 50+5(x-6) with a domain of {x | 7,8,...}. With this information, we can write our piecewise function.
M(x)= 50, & -3pt {x|1,2,...,6} 50+5(x-6), & -3pt {x|7,8,...}
b From the exercise, we know that per child Jill must pay $3 for party favors and treats. This gives us a second function, which is also defined for only natural numbers.
E(x)=3x, {x|1,2,3,...}
c The income that Jill makes, N(x), can be expressed as the difference between M(x) and E(x).
N(x)=M(x)-E(x)
d To create a function for N(x) we should subtract 3x from each of the parts in our piecewise function from Part A. For the first part we have 50-3x. Let's simplify the second part.
