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Functions on the Real Line - Example 4

In mathematics, a function (or map) f from a set X to a set Y is a rule which assigns to each element x of X a unique element y of Y, the value of f at x, such that the following conditions are met: 1) For every x in X there is exactly one y in Y, the value of f at x; 2) If x and y are in X, then f(x) = y; 3) If x and y are in X, then f(x) = f(y) implies x = y; 4) For every x in X, there exists a y in Y such that f(x) = y.


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Okay, so we're gonna find the domain in the range for this function here. So I'll tell you that the domain is typically easier to find and also more useful to find. The range is really interesting to find, but a lot of times in practice, it's not as necessary. But here the domain recall that we want the rial numbers for which we get a meaningful output. Or, in other words, the output is a real number. And here the Onley problem we have is if we had a negative under the square root. So for the domain, we want to require that's X plus two be bigger than or equal to zero, because I can on Lee, take the square root of a number that is bigger than or equal to zero. But that means that X has to be greater than or equal to negative, too. So then the domain of this function is just negative to to infinity, including negative too. Okay, so the range is a little bit trickier, and it requires maybe some clever observation. So the first one is that notice that f of X is always greater than or equal to zero so we might be suspicious that the range of f be 02 infinity. And now you say Okay, great. Well, we're done. Well, no, we're not, because this is really just a guess. I'm just assuming that. Well, okay. For every number between zero and infinity, I confined a number X Such a square root of X plus two is that number toe actually prove that the range is this set. I need to take a why in this set. So if I take a why here? And I said it equal to square root of X plus two. So this why is just gonna be a number between zero infinity? Well, if I square both sides, I get why squared is equal to X plus two. Or, in other words, X is equal to why squared minus tip. Okay. So why did I do this? Because what this allows me to do is see Okay for every Why between zero and infinity, I can actually find an input. What is the input? Why squared minus two? So that when I plugged that input in here, I'm actually going toe output. Why? And so I can actually conclude the range of dysfunction is 02 infinity, including zero. And here, of course, was the domain. Alright, so let's start with the domain and actually hear the domain is really easy. So when you're looking for the domain of a function, you're looking for things like square, it's and denominators. So can't have negatives under square roots, and you can't have zeros and denominators. Well, there's no square roots or denominators here, so the domain is actually all real numbers. There's no problem. We can plug in any number, any real number in for X, and we'll get a real numbers and output. Okay, so let's look at the ranch. So the range again, we're going to have to use some clever observations. And first of all, notice the X minus three squared is greater than or equal to zero. Okay. And if I subtract two, which is, of course, just f of X, so x minus three squared minus two. Well, I've just subtracted to from both sides of this inequality, so ffx appears to always be greater than equal to negative two. So we might guess that the range is going to be negative to to infinity, including negative, too, but we actually have to justify that. So if I pick a why in the range here between negative to infinity, I need to actually show that there is an input that gives me that output. Okay, so I'm picking a number. Why? Between negative to infinity. And so I want y to be X minus three squared, minus two so I can add to and then X minus three squared. I'm going to take the square root of both sides. So I have square root of lifeless too. Equals? Well, really. I'm going to get absolute value of X minus three, but I'm just looking for one X. So let's just take the positive square roots X minus three. But then look, as long as why is greater than or equal to negative two, this will be a real number. And I could take X to be square root of why, plus two plus three. So this input that I choose given any number, why here will give me an output of that. Why? When I plug it back into the function therefore their range will be Yes, sir. And we've already found the debate. Okay, so let's start with the domain. So notice that the only problem here is if the denominator is your So we're going to require that X squared minus X not be equal to zero. And so the way I'm going to solve this now this is sort of strange, but I can treat this just like inequality. But I'm just saying, Well, I just don't want it to be equal to zero, so I can factor. This is X times X minus one not being equal to zero. So now for this to not be zero, both of these can't be zero. So we better have The X is not zero. And then, of course, here X can't be one. So we have again. And both of these things need to be true. Ex cannot be won. So the domain well, we can just write. It is X not equal to zero and economical toe one. But just for fun. Let's write it an interval notation. So we'll have negative infinity all the way up to zero. Not including zero union 01 Not, of course not, including zero or one and then one off to infinity, not including one so there's the debate. So the range here is a little tricky. And again, you just have to kind of go with your gut. So let's suppose that we're looking for the range here and I want to notice right away something that's true. The ffx is never equal to zero. And why is that? Well, the only way that ffx is going to be zero is if the numerator is zero. But the only way the numerator is euro is it x zero. But X is not in the domain of the functions. F of X can never be zero. So I'm just going to make a guess. Now it's a very educated guess, and the educated guess is going to come in what we do next. Really, I'm I'm so I'm going to solve this problem by working backwards. But I'm going to guess that the range of this function is actually negative. Infinity 20 and then 02 Infinity. So basically every real number except zero. So let's say that we had a number. Why in this ranch? So I need to show actually that I confined an input given any. Why here? I confined an input that when I plug that input in, I get Why so I would like why to be X over X squared minus x So let me just switch some things around and so will have Why times x squared minus x And here what I'm doing is I'm just gonna cross multiply equals X And then So I'm trying to solve for X here. So this is why times X squared minus y X is equal to X and let me subtract X over. So why squared minus Why X minus X is equal to zero. Okay. And let's see, he has This is gonna just be why X squared. And then here will have why, plus one times X is equal to you. Zero. Okay, then we'll go up here. True. So this is definitely some algebra, and I'll go ahead and I wrote this before, but I'll say it now. The hardest part here is algebra. And so Okay, I wanna factor this. I'm gonna factor the excel. So I have an ex times. This will be wine Times X. And then what's left over here is minus. Why? Plus once it was minus, why minus one and so either X is zero. But remember that we're actually excluding X equals zero because it's not in the domain. So that means it must be the case that why x minus y minus one equals zero. Okay, so let's just solve for X. So this is going to be why X equals y plus one. And then I can divide by Why? So that X is equal to why plus one over why and notice that why is not zero. Why is a number other than zero? So as long as why is not zero? I actually get a meaningful input here, and that input is going to give me why when I plug it back into the function, Okay, so that proves my guess. And really, what I should say is that my guests came from first solving essentially for an inverse the function and seeing what inputs I need to give me specific happens. And, of course, here's the domain. So, like I said, finding the range is a little bit difficult. It's a little bit more theoretical, involves a little bit of proof technique that we're not going to get into the one in this class But that's how you do it. And then the domain again. We're just looking for no negatives, underscored roots and then, of course, no zeros in the denominator of fractions.