in this video i want to give you a basic introduction into domain and range and how to write the expression using interval notation given the graph of a function so what is the domain and range of this particular function now the domain tells you the x values of the function the range tells you the y values of the function so let's focus on the domain first what is the lowest x value of the function the lowest x value as seen here is negative four the highest x value is three and this function contains every x value in between negative four and three so in interval notation we could say that the domain is from negative four to three and we're going to use brackets because it includes negative four and three so that's how you can write the domain for this particular function now what about the range what's the range of this particular function well let's focus on the y values the lowest y value that we see this is negative five now the highest y value is four and there's no breaks in the graph this graph is continuous from negative five to four along the y axis so therefore the range is going to be negative 5 to 4 and so that's a simple way to determine the domain and range of a function using a graph now go ahead and try that example find the domain and range of the function so let's start with the x values the lowest x value is at negative six and the highest we could see is positive six now notice that we have an open circle at negative six so negative 6 is not included so i'm going to use the parentheses for that so it's going to be negative 6 to 6 but here we have a closed circle so that's going to be associated with a bracket and so that's the domain for this particular function now what about the range so let's focus on the y values the lowest y value occurs at negative 4 and the highest y value occurs at five but negative four is not included so it's going to be negative four to five but five is included so that's the range for this particular function go ahead and try this example what's the domain and range of that function so let's start with the x values the lowest x value is one now notice we have an arrow so this goes all the way to infinity so therefore the domain is going to be from 1 to infinity now how about the range the lowest y value is 2 and because of this arrow it's going to go up and to the right indefinitely so therefore the highest y value technically is infinity because this doesn't end so the range is going to be from 2 let me write that better 2 to infinity and 2 is included because we have a closed circle now here's the next example we have a downward parabola and how can we determine the domain and range of it so let's start with the x values so once again we have an arrow that means it's going to go down as it slowly goes to the left so it's going to keep going to the left forever and we have an error on the right side so it keeps going to the right side forever so therefore the lowest x value because it goes all the way to the left is negative infinity and because it goes all the way to the right the highest x value is positive infinity so the domain is going to be negative infinity to infinity always use a parenthesis symbol next to an infinity symbol now let's focus on the range the y values the lowest y value we can clearly see that it's a negative infinity because it keeps going down forever but the highest y value is stream and never goes beyond three so the range starting from the lowest value to the highest value is negative infinity to three including three and so that's how you can determine the domain and range of a parabola now let's try some harder examples if you want to pause the video and work on this one so let's start with the domain let's focus on the x values so the lowest x value is negative six the highest x value is five and notice that there's a jump in the graph at negative one however x can be negative one so x could be anything between negative six and five except negative six because we have an open circle at negative six but it can be negative one because we do have a closed circle at negative one so the domain for this one is going to be negative six to five because it could be any x value between negative six and five just not negative six now what about the range what about the y values so the lowest y value that we see is negative four and the highest is positive four now notice that there's nothing between negative one i mean negative two and one so y can't be anything between there it could be negative two though because we do have a closed circle at negative two but it can't be negative one negative point five point eight it can't even be one because we have an open circle at one so how can we describe the range using interval notation in this example what we need to do is we need to use the union symbol that will connect the range of this expression with the range of this expression omen and everything in between since there's nothing there so we're going to go in this direction from the low value to the high value so the lowest value the lowest y value that is is negative 4 and we need to use parentheses because we have an open circle so it goes from negative four to negative two now we have a closed circle at negative two so we're gonna use brackets and then union so this is for the first graph so let's connect it to the second part of the graph we're going to start back up at 1 and end at 4. now 1 is not included we have an open circle but we have a closed circle at 4 so 4 is included and so this represents the range of this particular function so hopefully this makes sense to you and the best way to learn this is to do a few examples so this is going to be the last example for this video go ahead and determine the domain and range let's see if you understand how to do it now so let's start with the x values the first x value of interest is at negative eight the next x value that i want to take note of which ends this portion of the graph that's that negative four and then the second part of the graph starts back up at negative two and it ends at five now the third part of the graph it starts back up again at seven and then we have an arrow so it goes to infinity so basically these numbers that we see here we just have to use that to write the domain so the lowest x value it starts at negative eight and it includes negative eight and then it stops at negative four but it doesn't include negative four so let's use parentheses union now for the second part of the graph it starts at negative two and it includes it and it stops at five and it doesn't include 5. and then union it starts back up at 7 and then ends at infinity so that's the domain for this particular graph now let's focus on the y values the range so let's focus on this one the lowest value that i see here is negative six now this is not the highest value of this function so i'm not going to worry about it the highest value is here which is that one so this graph includes everything from negative six to one notice that this open circle is not relevant it's not equal to negative 4 at this point but it is equal to negative 4 at this point if you draw a horizontal line notice that there's two possible locations at which it can equal negative 4 here and here it doesn't equal negative four here but it does equal negative four here so y can be negative four so we have everything from negative six to one now let's focus on this one i don't need to worry about this point because it's already included in this graph the highest y value here is two so notice that y could be anything from negative six to two it could be one here and here so i don't need to worry about this one now notice that there's nothing between two and five there's no graph in this region so that's where i need a union now the highest y value is going to be infinity because of the arrow so these are the points of interest so the range is going to be negative six to two it does include two and then union five it does include five to infinity now if this section confuses you here's what you can do now let's focus on these two parts separately let's say if we want to write the range of each one separately the first one just this portion is going to be negative six to one and it includes one now the second part is going to be this starts at negative three by the way and it stops at two now if we want to find the union between these two expressions or these two sets represented in interval notation what would it be the union of those two is going to be negative six to two let's say if we drew a number line here's negative six here's one and then for the second one this is going to be negative three and this is going to be two so here's the first one negative six to one and here's the second one negative three to two so if we combine those two into one number line if we found a union between them it's going to look something like this it's going to start at negative 6 in a different color and then it's going to stop at 2. which gives us this expression so what you could do is find the range for each portion of the graph separately and then just find a union for anywhere they overlap and that would still give you the range or if you could see it graphically i would just do it the first way but that's it for this video thanks again for watching you