Domain and range of lines, segments, and rays | Algebra 1 (TX TEKS) | Khan Academy

So what we have here is two different F of XS defined by their graphs, and what we want to 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. So actually, let's remind ourselves what a function is. A function can take an input X, and then it is, so 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 X's that this function can take? So 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. So what are all of the potential x's over which this function is defined? Well, it's not defined if x is equal to -6. I don't know what f of x is there. We can see it is defined at x at -4 because I see that f of -4 is 8 right over there, and I'm going to fill it in to show that it's definitely defined there.

It's defined from -4, and we can keep increasing x, keep increasing x all the way until, but not quite at 8, because right at x equals 8, the function isn't defined. We have an open circle there. It is 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 being -4.

So x is going to be -4 is less than or equal to x, which is then going to be less than 8. Why didn't I say less than or equal to 8? Because it is an open circle here. The function is not defined at x = 8.

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 at, it looks like at x equals 8. We don't quite take on, we're tempted to say that f of 8 is 2, 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 x's, we can see that our function is defined. And so the function can take on values right above 2, all the way to, it looks like the function can take on a value as high as 8. So I will circle that in right over here.

We can see that f of -4 is 8. So how would we define the range right over here? Well, we can start at 2, but the function can't take on 2. So the function is going to be greater than 2. It's not greater than or equal to, or 2 is less than the function.

It's not 2 is less than or equal to, but then the function can go all the way up to including 8. 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 in 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. So you could take any x right over here, and the function is defined.

I can tell you what f of -10 is; I could tell you because 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. So the range here is all real values of f of x, and we are done.