Function Composition Thinking Problem

Question

Here is the question: A banquet hall charges $\$975$ to rent a room, plus $\$39.95$ per person. Next month they will offer a $20\%$ discount off the total bill. Determine two equations, one for cost, one for the discounted bill. Then using composite functions, express the discounted cost as a function of the number of people attending.

So I know that the $\$39.95$ is going to be one of the variables. So I used $x$ for the number of people. I had $f(x)=975 + 39.95x$. However I am unsure about the second equation. I thought of like $g(x)=0.20(a)$ where $a$ is the total amount. However this will not work well with combining the functions. If someone could help me, I'd really appreciate it.

Answer 1

A statement like $g(x) = 0.20(a)$ is a bit of a mis-match: The notation $g(x)$ tells us the function should depend on $x$, but $0.20(a)$ (and your remarks) suggest that you're considering $0.20(a)$ to be a function of some variable $a$ (the amount of a bill, which is quite smart).

So, you have two options:

• Use $x$ instead of $a$, so that $g(x) = 0.20x$. This is really the same function you listed before, but it's a semantic difference about the name of the variable (here, we'd be thinking of $x$ still as the amount of some bill).

• Commit to using $a$ as your variable, and write your function as $g(a) = 0.20a$, where $a$ still stands for the amount of some bill, and $g(a)$ should let you know how much you'll be spending after the $20\%$ discount is applied.

That said... the function $g(a) = 0.20a$ actually tells you how much the discount is ($20\%$ of your bill, $a$), not how much you're paying after the discount. If you get $20\%$ off, you still have to pay $80\%$ of the price, so the function $g(a) = 0.80a$ is actually what you're looking for.

If instead of thinking of $a$ as the amount of a bill, but just some generic amount (of money, or people, or gallons of water, or...) then the function compositions $(f \circ g)(a) = f(g(a))$ and $(g \circ f)(x) = g(f(x))$ both make sense, mathematically. But if you stick to your guns and insist that $x$ should be some number of people, and $a$ should be the amount of some bill, this actually tells you exactly which composition you want (the two do not give the same function).

Answer 2

Let $x$ be the number of people. The cost of renting a room can be represented as a function of the number of people: $f(x) = 975 + 39.95x$.

Let $y$ be the amount of a bill. The cost after a 20% discount can be represented as a function of the amount of a bill: $g(y) = .8y$.

A 20% discount on the cost of renting a room can be represented by a composition of the two functions: $g(f(x)) = .8f(x) = .8(975 + 39.95x) = 780 + 31.96x$, which expresses the discounted bill as a function of the number of people, $x$.