A couple planning their wedding has found that the cost to...

A couple planning their wedding has found that the cost to hire a caterer for the reception depends on the number of guests attending. If 100 people attend, the cost per person will be 20. 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 (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? 95 attend, $x=5$

A couple planning their wedding has found that the cost to hire a caterer for the reception depends on the number of guests attending.
If 100 people attend, the cost per person will be 20. 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
(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?
95 attend, $x=5$

Key Concepts

Function Composition

The final total cost is expressed as a product of two functions: one giving the number of guests and the other giving the cost per guest. Function composition, or combining functions through operations like multiplication, is a fundamental concept in algebra that enables the modeling of more complex relationships by integrating simpler functions together.

Variable Representation and Function Definition

The use of variables, such as x representing the number of people not attending, is central to forming functions that model real-life situations. It highlights how variables can be defined in relation to a known quantity (here 100 guests) to build functions such as the number of guests attending, the cost per guest, and ultimately the total cost. This approach simplifies the process of defining related functions in a clear, systematic manner.

Translating Word Problems into Functions

This concept involves defining mathematical functions based on the relationships described in a real-world scenario. The problem gives a narrative where the number of guests and the corresponding cost per person change based on how many guests attend. Understanding this concept allows one to represent these variable relationships using functions, translating the situation into mathematical expressions that can be analyzed or computed.

Linear Relationships

In this problem, the cost per guest increases linearly as the number of guests decreases. Recognizing this linear behavior is key, as it allows for the use of linear functions to model the relationship between the number of guests who do not attend and the corresponding increase in cost per person. Linear functions are characterized by a constant rate of change, which is evident in the fixed increment of cost per each additional guest not attending.

Transcript

00:02 In this problem, we're looking at how to write a function given some information in a word problem.

00:08 And so in this problem, we're talking about people are trying to plan their wedding, and they're trying to figure out how much the caterer is going to cost.

00:15 The caterer said if there are 100 people at their wedding, then it's $20 per person, but every person less than 100, the cost will go up by $5.

00:24 So we're trying to assume no more than 100 people will be there, where x is going to represent how many people less than 100.

00:32 So like if 90 people come, x is 10 because that's 10 less than 100.

00:38 And so for part a, we need to write a function that we're going to call n of x, and that's going to represent how many guests will be at the wedding.

00:47 Okay, so if the maximum is 100, so we've got our function n of x, and this is part a, n of x is representing the number of people, and there's a maximum of 100 people, and then we're going to subtract x is how many people less than 100.

01:06 Okay, so if 90 people were going to be there, x would be 10, 100 minus 10 would give us that 90.

01:12 So this function here represents the number of people that are attending the wedding.

01:19 And then for part b of this problem, part b says, we need to now write a function g of x that gives us the cost per guest.

01:28 And so the problem said that every person less than, 100 is going to make the is going to make the price increase by five by five dollars so for our function if we're calling this one g of x it originally if there's 100 people it's going to cost 20 dollars per person but any number of people less than a hundred we're going to have to add on five dollars more per person that there is less than a hundred so g of x the cost per person is $20 plus five times however many people less than 100.

02:04 So, for example, if there was 99 people, then x is one.

02:09 So there's one less than 100.

02:10 So that means it would cost 25 per person.

02:12 So we add $5 on for every extra person underneath.

02:16 That's less than 100.

02:18 And part c is asking to write a function that gives us the total cost, which they already told us would be n of x times g of x.

02:26 So part c, we're calling our cost function is going to be c of x, which they told us is just n of x times g of x.

02:44 And we've already got our functions for n and g...