Question

domain & range quiz

1. imagine a coffee shop that offers catering. they charge a $20 delivery fee and an additional $6 for each coffee served.

a) write a function for the total cost, c, in terms of n, the number of coffees served. (1 pt)
b) what is the domain and range of the function? (1 pt each)

1. a company that prints custom shirts charges a one - time setup fee of $25 and an additional $8 per shirt printed. the total cost is determined by the function c(x)=8x + 25, where x is the number of shirts ordered.

a) find and interpret c(9) and explain what it means in context of the question. (1 pt each)
b) c(39)=337. explain what it means in context of the question. (1 pt each)
c) is this a discrete or continuous function? explain. (1 pt)

Explanation:

Step1: Write cost - function for coffee - shop

The coffee shop has a $20 delivery fee and $6 per coffee. So the cost function $C(n)$ (where $n$ is the number of coffees) is a linear function of the form $y = mx + b$. Here, the slope $m = 6$ (cost per coffee) and the y - intercept $b = 20$ (delivery fee). So $C(n)=6n + 20$.

Step2: Find domain of coffee - shop function

The number of coffees $n$ cannot be negative. In a real - world context, $n$ is a non - negative integer (you can't serve a fraction of a coffee in this context). So the domain is $\{n\in\mathbb{Z}:n\geq0\}$.

Step3: Find range of coffee - shop function

When $n = 0$, $C(0)=20$. As $n$ increases, $C(n)$ also increases. Since $n$ is a non - negative integer, $C(n)=6n + 20$ where $n\in\mathbb{Z},n\geq0$. So the range is $\{C(n)\in\mathbb{Z}:C(n)\geq20\}$.

Step4: Evaluate $C(9)$ for shirt - printing function

Given $C(x)=8x + 25$, substitute $x = 9$ into the function. $C(9)=8\times9+25=72 + 25=97$. In the context of the shirt - printing company, it means that when 9 shirts are ordered, the total cost is $97$ (the sum of the $25$ set - up fee and $8$ per shirt for 9 shirts).

Step5: Interpret $C(39)$ for shirt - printing function

Given $C(x)=8x + 25$, when $x = 39$, $C(39)=8\times39+25=312+25 = 337$. It means that when 39 shirts are ordered, the total cost is $337$ (the set - up fee of $25$ plus $8$ per shirt for 39 shirts).

Step6: Determine type of shirt - printing function

The number of shirts $x$ can only be non - negative integers (you can't order a fraction of a shirt). A discrete function has a countable number of distinct values. Since the input values (number of shirts) are non - negative integers, the function $C(x)=8x + 25$ is a discrete function.

Answer:

1.

• a. $C(n)=6n + 20$
• b. Domain: $\{n\in\mathbb{Z}:n\geq0\}$, Range: $\{C(n)\in\mathbb{Z}:C(n)\geq20\}$

2.

• a. $C(9)=97$. It means the total cost of printing 9 shirts is $97$ (including the set - up fee and per - shirt cost).
• b. $C(39)=337$. It means the total cost of printing 39 shirts is $337$ (including the set - up fee and per - shirt cost).
• c. Discrete function. Because the number of shirts printed $x$ is a non - negative integer.