Domain and range from graphs of quadratic and exponential functions | Khan Academy

So what we want to do in this video is try to figure out the domain and ranges of G of X that's depicted right over here and H of X that's depicted over here. So pause this video and see if you can figure out the domain and range of each of these functions.

All right, now let's start with G of X. Now, the domain, just as a reminder, this is all of the X values over which G of X is defined. Or if we view a function as you can take some input and you put it into that function and you're going to get an output, which in this case would be G of X. Our domain is what are all of the things that I could input here?

We could see that this graph keeps going to the left; it keeps going to the right. So I could input any real number over here for X. So the domain here is all real values of X.

Now what about range? So let me write that over here. Range: these are all of the possible values that our actual function could take on. So what's the highest value that our function can take on? Well, the highest value for this graph right over here looks like F of X is equal to two.

And then every other value that it takes on is lower than that, and it seems like it can take on an arbitrarily low value because this function keeps decreasing on either side. So the range looks like it maxes out at F of X is equal to 2.

So another way to think about it is F of X is going to be less than or equal to two. That's all the possible outputs right over here. Now let's do this function H of X. So what are all of the X values it could take on? Well, this one isn't defined for all X's.

So it looks like this is halfway between zero and two. So this right over here looks like it's at one, but it's an open circle, so it's not quite defined at one. But as soon as we get less than one, it is defined. Any of these values I can take an X and figure out its F of X, and it seems like I can just keep going lower and lower and lower.

Because even though this function, as you go to the left, it looks like it's increasing very quickly, but it just keeps going to the left. So I can have any X value that is less than one; it seems like H is defined there.

So our domain in this situation, I'll write it up here: the domain is it's defined for any X less than one. Any X less than one. And then what is the range? What is the range?

Well, it looks like right at one it's not defined; it's not defined. But for any higher than that, the function can take on that value. Or another way I could say it is at one, the function can't quite take on that value; it doesn't look like it can, but anything higher than that, it can.

So it looks like H of X, H of X is greater than one, can take on any value larger than one, and we are done.