In math there is not just one kind of function, or formula, but many. There are different functions for different situations. But let's start with the notion of a function. A function does something like this: if I'm at the gas station to buy gas, then two numbers become relevant, and they are (a) the number of gallons of gas that get put into my car and (b) the dollar amount that I have to pay. Now, you can get many different amounts of gas, and pay many different prices. But, there is a definite order and predictability to the whole situation, and a mathematical function does the job of telling us how.
Now, about this function. For the situation at the gas pump, the simplest (in my opinion) type of function does the job. It's called a linear function. Before I get into that, let me state that I realize most people understand how to buy gas and most people probably don't know what a linear function is. And so that may seem to make a discussion of the function unnecessary? All I can ask is to hear me out. Right now gas is about $3 per gallon. As I said, people know what this means: 1 gallon costs you $3, 2 gallons costs $6. We could really get complicated and say that 2.5 gallons will cost $7.50. But to just list numbers like that, although it is correct, does not reach for the whole truth of the situation. What I'd like to do is describe what happens to my bill as I put more gas in the tank. The more gas I buy, the more I pay, and the general truth is that for each additional gallon of gas I buy, I will pay an extra $3. The truth I want to capture here is that no matter how much gas is already in the tank, the next gallon I put in will cost an extra $3. This being the case is what makes the function that determines my bill for buying gas a linear function.
Now, I'd put money on it that most people would accuse me at this point of providing unnecessary details for something already sufficiently understood. But that may be because they think the mathematics of all situations are like the aforementioned one about buying gas. True, many common ones are (for the most part), such as getting paid by commission, or getting paid by the hour, or paying taxes. The explanations for these examples would be of the same nature as that of buying gas. If you get paid $10 per hour, then every extra hour you work gets you $10 more dollars, regardless of how many hours you have worked already (barring complications, such as overtime). This is the essence of a linear function: two quantities (i.e., gas and money, time and money, etc.) are linearly related if there is a specific change in the one that always produces the same change in the other. But now I offer you to consider that not all situations call for linear functions.
To begin to get a grasp of how a situation could be unsuitable for a linear function, consider the population of a city at any given time. Here, various moments in time will correspond to various levels of population in the city, so one may think there is a function that explains how. First, we can probably all agree that as time goes on, the population gets bigger. But is it linear? For it to be linear, something like this would have to be true: for every 20 years that goes by, the population will increase by 750 people. Now, let me see if I can convince you that population growth in general will probably not be linear. Consider for the sake of argument that the original population was 500, and then 20 years later it was 1250 (or it grew by 750). Just like in the gasoline example, we have to start by agreeing on our the mechanics of example. So, let's assume something like this happened: all couples will have 3 children and they all live a very, very long time. Of the 500 original people, there were 250 couples who each has 3 children, which is 750 children total, and a new population of 1250. But what will happen over the next 20 years? Will the population increase again by the same amount, 750 people? No, if the 750 new children, form 375 couples and each has 3 children over the next 20 year period, then there will be an additional 1125 new people and the new population will be 2375. In other words, the population grows by 1125 people during the 2nd 20 year period.
The point here is that the way the population grows cannot be linear: with my assumptions, it cannot grow by the same amount every 20 years. The general reason for this is that there is a repeating cycle that each reproductive group creates a new generation larger than itself. Now, you can argue with my assumptions that nobody died, or that there were no singles, or that all couples had children, or that all couples had 3 children. I concede that my assumptions are easily see to not true. But, it turns out that once all those complicating factors are considered, and we look at the raw numbers, populations usually grow in a way that is as if all new children form couples and then have a certain number of new children, and so on. This is commonly what happens, and population growth is commonly not linear. The type of function that describes populations such as this one is called an exponential function.
Before I go to the next type of function, I want to emphasize that in each situation I have been describing the effects of change: what happens as you put more gas in the tank, or what happens as time goes on. I'll start this last example the same way. Consider the perfectly reasonable situation that I am having new floors installed in my house. Suppose that each man that works on the floors can install 50 square feet per hour, and I have 2000 square feet for the new floor to go. Doing a little dividing will yield that one man can finish the job in 40 hours. It would be reasonable to think then that 2 men can finish the job in 20 hours, since two men would seemingly accomplish 100 feet in one hour. What happens if we push this further? Can 10 men finish in only 4 hours? Maybe. If we follow the math to its logical conclusions then 100 men can finish the job in only 24 minutes!
This is where I stray a little from my earlier examples. The math on its own is correct is saying that 100 men can finish the job in only 24 minutes, but practical considerations will prevent this. 2 men may not work individually at the same rate as one man will. And so on with 100 men. The reason involves things such as that there are limitations on tools available and space. Will 100 men on the floor, the crowded conditions alone will prevent them from individually working at the same rate of any one man working alone. We might then grant some reality to this assumption: as more men work on installing the floors, then the effect will they will individually work more slowly. To give some numbers, it may work something like this: one man working alone can work at 50 square feet per hour, but for each additional man they will each work at 10 square feet per hour slower. Now, if we are concerned with how the number of workers affects the time it takes to finish the project, then what kind of function will we get? It turns out that the decreased productivity caused by additional workers here leads to a function that's not linear or exponential. Doing some math will give these results: 1 man finishes in 40 hours, 2 men will finish is 25 hours, and 3 men will finish in 22.2 hours. So it seems that more men means less time to finish, despite decreasing individual productivity. But as we keep going, we find that 4 men will take 25 hours, and 5 men will take 4 hours. So, as we add more men to the job, the time to finish decreases to a minimum and then increases thereafter. That's what's unique to this function, and it's called a quadratic function.
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