Topic A: Functions
We're asked to look at the table below. From the information given, is there a functional relationship between each person and his or her height? So a good place to start is just think about what a functional relationship means. Now, there's definitely a relationship. They say, hey, if you're Joelle, you're 5-6. If you're Nathan, you're 4-11. If you're Stewart, you're 5-11. That is a relationship. Now, in order for it to be a functional relationship, for every instance or every example of the independent variable, you can only have one example of the value of the function for it. So if you say if this is a height function, in order for this to be a functional relationship, no matter whose name you put inside of the height function, you need to only be able to get one value. If there were two values associated with one person's name, it would not be a functional relationship. So if I were to ask you what is the height of Nathan? Well, you'd look at the table and say, well, Nathan's height is 4 foot 11. There are not two heights for Nathan. There is only one height. And for any one of these people that we can input into the function, there's only one height associated with them, so it is a functional relationship. We can even see that on a graph. Let me graph that out for you. Let's see, the highest height here is 6 foot 1. So if we start off with one foot, two feet, three feet, four feet, five feet, and six feet. And then if I were to plot the different names, the different people that I could put into our height function, we have-- I'll just put the first letters of their names. We have Joelle, we have Nathan, we have Stewart, we have LJ, and then we have Tariq right there. So lets plot them. So you have Joelle, Joelle's height is 5-6, so 5-6 is right about there. Then you have Nathan. Let me do it in a different color. Nathan's height is 4-11. We will plot to him right over there. Then you have Stewart. Stewart's height is 5-11. He is pretty close to six feet. So Stewart's height-- I made him like six feet; let me make it a little lower-- is 5-11. Then you have LJ. LJ's height is 5-6. So you have two people with a height of 5-6, but that's OK, as long as for each person you only have one height. And then finally, Tariq is 6 foot 1. He's the tallest guy here. Tariq is right up here at 6 foot 1. So notice, for any one of the inputs into our function, we only have one value, so this is a functional relationship. Now, you might say OK, well, isn't everything a functional relationship? No! If I gave you the situation, if I also wrote here-- let's say the table was like this and I also wrote that Stewart is 5 foot 3 inches. If this was our table, then we would no longer have a functional relationship because for the input of Stewart, we would have two different values. If we were to graph this, we have Stewart here at 5-11, and then all of a sudden, we would also have Stewart at 5-3. Now, this doesn't make a lot of sense, so we would plot it right over here. So for Stewart, you would have two values, and so this wouldn't be a valid functional relationship because you wouldn't know what value to give if you were to take the height of Stewart. In order for this to be a function, there can only be one value for this. You don't know in this situation when I add this, whether it's 5-3 or 5-11. Now, this wasn't the case, so that isn't there and so we know that the height of Stewart is 5-11 and this is a functional relationship. I think to some level, it might be confusing, because it's such a simple idea. Each of these values can only have one height associated with it. That's what makes it a function. If you had more than one height associated with it, it would not be a function.