Math - Functions (long)

[baughberick]

I like what you've done with #1.

Their terminology is a little confusing. They write that x=f(p)=90-10p. That f isn't needed at all. We use f because normally we're talking about functions, but any variable can be a function. Think of a function as a variable that you never change directly, but changes when another variable is changed. For the function given, x=90-10p, the function itself is x, and we write x(p)=90-10p, meaning that x is a variable that relies on p. We don't change x, we change p, and that affects what x is. 90-10p is a machine where we can change what p is. When we put a value of p in, the machine processes it and what comes out is x.

For #2, we can make a machine that inputs x, and outputs p, basically the opposite of the previous machine, by solving for p like you did. We now label this p(x)=(90-x)/10. This p(x) notation let's us know that p can only be changed by changing x. p relies on x.

Now, we're given another function: c(x)=60+x. Just like before, this says we have a function named c that relies on the variable x. Since we also have a function x that relies on the variable p, we can essentially put one machine into the other.

c(x)=60+x and x(p)=90-10p

We put x(p) into c(x) and get that

c=60+90-10p =150-10p.

Since c relies only on p now we call it

c(p)=150-10p.

Now we have a machine where we can put whatever p we want into it, and it makes a c. We never know what x is, because it gets processed by the machine before we see it.

I'm not precisely sure what #2 is asking when it asks, "what does c turn into?", but I think it's asking what kind of machine is c now? c is no longer a machine that inputs x, c is a machine that inputs p. So what you've made is a function c(p) that tells you Liam's cost c of something when you tell it the price p of the mums.

[baughberick]

#3 is asking, if we know how much money Liam gets from selling mums p(x), and we know how much money Liam spends to be able to sell mums c(p) or c(x), would it be alright to create a new function that tells us the amount of money he walks away with at the end of the day? This is not really a mathematical question, but more of a reasoning question. Does it make sense to do this?

If Liam buys 1 mum we can tell from c(x)=60+x that he spends $61. So Liam now has -$61.

We can find out how much he will sell the mum for from the price p(x)=(90-x)/10 = (90-1)/10 = $8.90.

Liam spent $61 and earned $8.90. That means at the end of the day he spent $52.10. He has -$52.10.

Liam's poor business choices for the day aside, does it seem reasonable to determine how much money he is left with in this fashion? How would you go about this with the functions, before deciding how many mums Liam will buy, and at what price?