Solve each problem. The cost to hire a caterer for a party...

Solve each problem. 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 .$ 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, $x=5$ (a) Write a function defined by $N(x)$ giving the number of guests. (b) Write a function defined by $G(x)$ giving the cost per guest. (c) Write a function defined by $N(x) \cdot G(x)$ for the total cost, $C(x)$ (d) What is the total cost if 80 people attend?

Solve each problem.
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 .$ 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, $x=5$
(a) Write a function defined by $N(x)$ giving the number of guests.
(b) Write a function defined by $G(x)$ giving the cost per guest.
(c) Write a function defined by $N(x) \cdot G(x)$ for the total cost, $C(x)$
(d) What is the total cost if 80 people attend?

Key Concepts

Substitution and Evaluation

Substitution is the process of replacing a variable in a function with a specific value to evaluate the function at that point. This is essential for determining outputs under specific conditions, such as calculating the total cost when exactly 80 people attend. It reinforces the concept of using mathematical expressions to make predictions or solve problems after modeling the situation.

Function Definitions

A function is a relationship that uniquely associates members of one set with members of another set. In this context, functions are used to model how the number of guests and the cost per guest change as the number of absences, represented by x, varies from its base condition. Understanding functions and their notation is fundamental for mapping real-world relationships into mathematical expressions.

Linear Functions

Linear functions express relationships with a constant rate of change and are typically written in the form f(x) = mx + b. In this problem, both the reduction in guest count and the increase in cost per guest are described by linear relationships, making it necessary to understand how changes in x affect the overall cost.

Mathematical Modeling

Mathematical modeling involves translating real-world situations into mathematical expressions or equations. In the given scenario, the problem models the cost structure of a caterer based on the number of attendees and the attendant cost adjustments. This process requires setting up functions that accurately represent the provided conditions and constraints.

Composition of Functions

The composition of functions involves combining two or more functions such that the output of one function becomes the input of another. Here, calculating the total cost involves multiplying the function for the number of guests by the function for the cost per guest, effectively composing them. This step is crucial for understanding how individual components interact to form the final model.

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