Domain and range of lines, segments, and rays

0:00- [Instructor] So what we have here is two different F of Xs defined by their graphs. And what we wanna do is figure out the domain and the range for each of these functions. So pause this video and try to figure that on your own before we do that together. Now, let's just remind ourselves what domain and range is.

0:18So actually, let's remind ourselves what a function is. A function can take an input X, and then it is, so they have a function F, and it's going to output something. It's going to output F of X. So a question is what are all of the Xs that this function can take?

0:34So the set of all things that this function can take in, all of the inputs that it can take in, that is our domain, and then all of the possible outputs, that is our range. So let's first think about the domain, and maybe I'll do this in different colors. So first, the domain.

0:51So what are all of the potential Xs over which this function is defined? Well, it's not defined if X is equal to negative six, I don't know what F of X is there. We can see it is defined at x equals negative four, 'cause I see that F of negative four is eight right over there, and I'm gonna fill it in to show that it's definitely defined there, and it's defined from negative four, and we can keep increasing X, keep increasing X all the way until, but not quite at eight, because right at X equals eight, the function isn't defined.

1:24We have an open circle there. It's defined everywhere up to that, but not including. Just as a reminder, the open circle means you can get close to it, but you don't include that number, while the closed circle means that you do include it. So what we can see is the smallest value in our domain is X is being negative four, so X is going to negative four is less than or equal to X, which is then going to be less than eight.

1:53Why didn't I say less than or equal to eight? Because it is an open circle here. The function is not defined at X equals eight. Now, what is the range going to be? The range is all of the potential values that the output, that F of X can take on. So we can start down here at X, it looks like at X equals eight, we don't quite take on.

2:19We're tempted to say that F of eight is two, but it's not. It doesn't quite count because we have an open circle there, but I'll put an open circle here because as soon as you get lower Xs, we can see that our function is defined, and so the function can take on values right above two, all the way to, it looks like the function can take on a value as high as eight, so I will circle that in right over here.

2:44We can see that F of negative four is eight, so how would we define the range right over here? Well, we can start at two, but the function can't take on two, so the function is going to be greater than two. It's not greater than or equal to, our two is less than the function.

3:03It's not two is less than or equal to, but then the function can go all the way up to including eight. And so we're done with the range of this first function. Now, let's do the same thing over here. What is the domain? I'll do that same purple color. Domain, well, it looks like pretty much any real number X that you were to input, the function is defined over it.

3:31So you could take any X right over here and the function is defined. I can tell you what F of negative 10 is. I could tell you 'cause this line just keeps going on and on and on. And so the domain is all real values of X, and now you could imagine what the range might be, because this line is going to keep increasing and increasing and increasing forever, so you can have an arbitrarily high F of X, and similarly, this line is going to keep decreasing and decreasing and decreasing forever as we go to the left, and so you could have an arbitrary low value of F of X.

4:15So the range here is all real values of F of X, and we are done.