1. First we are going to find the cost function.
We know for our problem that there is a 7% sales tax applied to the cost of her meal and she plans to leave 3$ as tip. Let [tex]c[/tex] represent the cost of the meal:
[tex]f(c)=c+0.07c+3[/tex]
[tex]f(c)=1.07c+3[/tex]
where
[tex]c[/tex] is the cost of the entree
[tex]f(c)[/tex] is the total cost she will pay
We know that the minimum cost of an entree is $5.98, so the domain of our functions so far is:Â [tex]5.98 \leq c \leq [/tex]
Now, to find the upper limit of the domain, we are going to take advantage of the fact the she only has $15 to spend, so we can replace the total cost with 15 and solve for [tex]c[/tex]:
[tex]f(c)=1.07c+3[/tex]
[tex]15=1.07c+3[/tex]
[tex]1.07c=12[/tex]
[tex]c= \frac{12}{1.07} [/tex]
[tex]c=11.21[/tex]
We can conclude that the domain of the function is [tex]5.98 \leq c \leq 11.21[/tex]
2. We know that Mia's friend Clara loans her $5, so Mia's total money now is $15+$5=$20. Since the minimum cost for an entree remains the same ($5.98), the lower limit of our domain remains the same: [tex]5.98 \leq c \leq [/tex]
Now, to find the upper limit of our domain, we are going to replace the total cost with 20 an solve for [tex]c[/tex]:
[tex]f(c)=1.07c+3[/tex]
[tex]20=1.07c+3[/tex]
[tex]1.07c=17[/tex]
[tex]c= \frac{17}{1.07} [/tex]
[tex]c=15.88[/tex]
We can conclude that the domain of the function representing the price of an entree Mia can afford is now [tex]5.98 \leq c \leq 15.88[/tex]
