 This video will talk about finding domain and range, specifically looking at graphs and being able to find the domain and range. But first we need to know what a domain and a range is. So the domain is all possible inputs. Usually that'll be our X's. What X's are allowed for this on this graph. So all possible inputs. And then it would probably make sense then that the range is going to be all possible outputs. Or typically your Y values outputs. And notice that it says possible. We could pick any X we want but some of them may not work. Or we could pick an X and then we find out that oh it doesn't have a Y. So then it would not have a possible output. Let's look at our first example here. This is a line. It's a linear function. So looking graphically, this line is going to go on forever up and forever down. If I had enough space on my graph, if I looked at like X is equal to negative 5, I would eventually hit my graph. And if I picked 5, I can see that one. If I picked negative 10 out here, then if I went down I'd eventually hit my line. So if you think about it that way, there are no X's that will not have a Y value. We would say that this is all reels. Every X I can plug into a linear function and get an output. And the shorthand writing for reels is a double backed R. So you can use either one of those and in fact if we talk about interval notation, interval notation would go from negative infinity to infinity. Interval notation uses parentheses and brackets and infinities always use parentheses. Because remember out here is going to be negative infinity on my X axis and over here will be positive infinity eventually on my X axis. So let's talk about the range then. How far up and how far down. Domain is how far to the left and how far to the right. Range is how far up and how far down does this graph go. And notice there are arrow heads on the ends of my line meaning it's going to keep going up forever and it's going to keep going down forever. So again we have all reels or double backed R or an interval notation and let me write that here so that we get the difference. Interval notation would be negative infinity to infinity. Let's look at this example. This is a quadratic that we'll eventually talk about. But quadratics are a little bit different than linear's in that you can see that we have a lowest point on this graph. Now again this graph is going to go up and it's going out as it goes up. So it's going out and up forever. So that means that if I came over here to 10 I eventually would go up and hit my parabola. It might be a really large number but I'd hit my parabola. So we would say that our domain has again all reels or negative infinity to infinity. And if we look at the range then we're thinking how high and how low does this graph go. Well we already know it's going to go up to positive infinity. It's going to go up forever. But it has a lowest point on it. It doesn't go down forever. It stops when y is equal to negative 4. So that means that we could say either y is greater than or equal to because it's right there on that point greater than or equal to negative 4 in interval notation we could write this. Smallest value is negative 4. The largest value is infinity. Negative 4 is included so we use a bracket in parentheses for infinity. This is what we call an exponential function and let's look at its domain and range. How far to the left and how far to the right. Once again this graph is going to go up forever and it continues to move out and this one's going to go down this direction and continue on forever and ever and ever. Now when we think about the domain how far to the left is it going to go? It's going to go to negative infinity. And how far to the right is it going to go? It's going to eventually go to infinity. If I went way out here over here and tried to go to my graph I might have to get to the ceiling but I eventually would hit my graph. So again we say that the domain is all reels. I'll put them all three up there again this time. All reels or the short hand is a double back bar. And if we do interval notation it goes from negative infinity the smallest value to infinity the largest value. Both get parentheses because they're infinity. Now the range is a little tricky here. When we think about the range we can see that it's going to go up forever. So it's going to go to positive infinity but it doesn't go down forever. In fact it's a little hard to see with this graph because we can't get it real fine here but you can see that this graph looks like it gets right on the x-axis but in reality it gets really close to it but never ever really crosses it. I hope you can see that it doesn't go below the x-axis. So we would say that this one and we're just writing what the values are we could say that y had to be greater than zero. When we talk about these functions it'll make more sense why it can't be equal to zero. It has to be greater than zero in this case. And in interval notation we would say that it goes from zero to infinity. It doesn't include zero because it's greater than zero so a parenthesis for zero and a parenthesis for positive infinity. So our final example here we're going to talk about domain and range of a model. And in a model you've got real life situation. If you see this graph I've drawn this graph so that you could see what the function of just the equation would be. But in real life we don't start with negative time back here so we start at this point right here and this is the path of a rocket and the ground is considered the x-axis. So we want to stop at the ground so we won't go below the x-axis at all. So in our domain we've gone from zero when time begins over to eight and we'll call this seconds and we'll call the y-value we'll call that feet. So when we think about the domain here we would say that it starts at zero seconds that's the smallest value and it ends at eight seconds and it starts at exactly zero seconds so that's a bracket and it ends at exactly eight seconds so it would be a bracket there. It includes those values. So for the range values then we can see that the lowest place that it hits is the ground and the highest point it gets to would be this point right here called the vertex that gets right straight across from 25 so the lowest value for the range would be the ground or zero feet and the highest point this toy rocket gets is 25 feet and it includes both. It hits the ground so it's included and it actually gets to 25 feet so that's also included. That would be the interval notation that's usually the easiest way to do model domains and ranges because it's definitely just an interval and that's how you find domain and range from a graph.