The cost to hire a caterer for a party depends on the number...

The cost to hire a caterer for a party depends on the number of guests attending. If 100 people attend, the cost per person will be 20 dollars. For each person less than $100,$ the cost will increase by 5 dollars. Assume that no more than 100 people will attend. Let $x$ represent the number less than 100 who do not attend. For example, if 95 attend, then $x=5$ (a) Write a function $N(x)$ giving the number of guests. (b) Write a function $G(x)$ giving the cost per guest. (c) Write a function $N(x) \cdot G(x)$ for the total $\operatorname{cost}, C(x)$ (d) What is the total cost if 80 people attend?

The cost to hire a caterer for a party depends on the number of guests attending. If 100 people attend, the cost per person will be 20 dollars. For each person less than $100,$ the cost will increase by 5 dollars. Assume that no more than 100 people will attend. Let $x$ represent the number less than 100 who do not attend. For example, if 95 attend, then $x=5$
(a) Write a function $N(x)$ giving the number of guests.
(b) Write a function $G(x)$ giving the cost per guest.
(c) Write a function $N(x) \cdot G(x)$ for the total $\operatorname{cost}, C(x)$
(d) What is the total cost if 80 people attend?

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Key Concepts

Linear Functions

Linear functions express relationships where change occurs at a constant rate, typically written in the form f(x) = mx + b. They model real world situations like cost changes or attendance reductions, where incremental changes produce proportional effects.

Function Modeling

Function modeling involves translating words and real-world conditions into mathematical functions. In this context, it requires setting up functions to depict the relationship between the number of guests, the cost per guest, and the total cost.

Function Composition

Function composition in this situation refers to constructing a total cost function by multiplying the function for the number of guests by the function for the cost per guest. It allows you to understand how two interdependent variables combine to form a new function.

Variable Substitution

Variable substitution is the process of replacing a variable with a specific value to calculate a numerical result. It is often used after developing a general mathematical model to compute outcomes for particular cases.

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