When getting into the higher levels of mathematics, it is necessary to completely understand what a function is and how you can use it. This blog entry will discuss this fairly simple topic and attempt to explain it in such a way that someone not versed in mathematics might understand it.

By the way: If you don’t want to read all this, click here to just watch a video…

Before we get started on what a function is, I wanted to discuss very quickly an example of why functions are important in Physics, Chemistry and yes, even business. Lets say that you are given the task of watching someone drop a baseball off of a very tall building about 450m tall ( yes very exciting! ). Your task is to sit and write down height values for every 1 second of its fall.

Since you are very board and aim to please, you quickly pull out your pencil and you start plotting points. These points might look like the following graph.

Note that in the above graph, we’ve plotted our points as the blue dots but there is also a red curve connecting them all. If you also refer to the left margin of the graph, you’ll also notice that there is a function there of 449.36+.96t-4.90t^2. So what does all this mean? Well lets dissect:

First of all, the dots represent a real world example of storing data. At some point however, people got tired of sharing all this data back and forth and started looking for other ways to represent the same facts. I mean, if we could see the same graph with a short function, who wants to sit and take a bunch of numbers and re-plot them to see what is going on? So, a function is nothing more than a mathematical representation of a path. At some point, someone was looking at all those points and said, hey Honey, take a look! If I take the number 1 and plug it into this formula, it puts a point in the exact same place as my data point. And a 2 does as well. So now, when Joe calls I can just give him this function and tell him to pick a point in time, plug it into the function and out comes the distance above the ground! She was very impressed.. :-)

So, a function is nothing more than a formula of sorts, where you plug in an x value and outcomes our y value. Just an easier way to deal with sometimes complex sets of data. However, in order to call your new function a function you must first test it by using the vertical line test. If you plot your points and at any time a vertical line dissects the graph in more than one place then you don’t actually have a function.

Also note that there are two key terms associated with the function. Domain and Range, in which the Domain consists of your x-values and the range consists of all the possible outputs of your y-value. So for example, in our case of dropping something off of the building that is 450m tall, a value of 500m for the range wouldn’t make any sense. At the same time, if it took 9.8s to fall to the ground, looking at a time of 15s wouldn’t make any sense either. So we’d say that our domain (our x value) consists of all points between 0s and 9.8s. Now, if we were to plug in 0 into our function it would give us 450m, and plugging in 9.8 would give us 0 height. So we’d say that our domain values that are valid are between 0 and 9.8.

Obviously, our original plot graph consisted of you taking 10 readings as the object fell to the ground. What happens tomorrow if I were to come to you and tell you that I need to know exactly where the object is at 1/2s intervals? Well, you’d have to get out your pad, run back to the building and drop another object so you could collect more detailed points to provide to me. What fun!

Functions to the rescue! What if I told you that I could give you a function to represent that objects path to the ground and you didn’t need any data points at all? And no matter what time interval you plug in, it will tell you precisely how high above the ground it was? Well, if you aren’t crazy over all this data collecting, you’d probably be very happy. No more collecting data, we have a FUNCTION!

We now know what a function is and I suggest you make sure that you understand the concept completely for a number of reasons. 1) its absolutely the simplest concept you’ll deal with in Calculus and 2) Its a necessity to understanding anything else I talk about in future discussions of Calculus.

–glenn hancock