in this video we're going to focus on relations and functions so what is a relation a relation is a set of pairs of input and output values here we have three ordered pairs in the first relation on the left the x value is the input value the y value is the output value the x values is associated with the domain of the relation the y values is associated with the range of the relation so now let's focus on part a list the domain and range of each relation so let's start with the domain so what we're going to do is we're going to make a list of all of the x values and i'm going to write it in ascendant order so first we have negative three and then zero and two now let's write the range of that relation so we're going to focus on the y values and it's already listed in ascendant order so 1 4 and 5. now let's do the same thing for the other relation so let's write out the domain so the lowest x value is negative two next is one and then three now let's write out the range of that relation the lowest y value is negative two and then it's three four and seven so that's how you can write out the domain and range of each relation now how can we determine if the relation is a function in order for the relation to be a function every input value must have only one output value if an input value corresponds to two or more output values that relation is not a function now let's focus on the first relation so we have the ordered pair 2 1 the input value is 2 the output value is 1. and then negative 3 4. so negative 3 corresponds to 4 and then 0 corresponds to 5. so for the first relation we can see that for every input value there's only one output value now let's focus on the second relation so we have the ordered pair one comma three next is negative two four and then it's three negative two and then finally negative two seven so for the second relation notice that negative two corresponds to two different output values now that's a problem if you put in an input value of negative two should the output be four or seven so whenever you have that situation you know that relation is not a function the first one is a function every input value corresponds to an output value just one output value so a quick way to look at a relation to see if it's if it's not going to be a function look for repeating x values if you see two x values that are the same but correspond to two different y values then you know the relation is not a function i want to take a minute to talk about my website video dash tutor.net it's a very simple website not too complicated but for those of you who want to be notified anytime i release specialized content in the form of a video an ebook an article it could be a digital course or podcast if you want to be notified feel free to join the email list and once you confirm your email you're going to get access to a page that has all of my playlists listed on it and this includes my final exam videos and also my test prep videos so feel free to join the email list when you get a chance and let's get back to the video now let's move on to the next example draw a mapping diagram of each relation shown below so let's start with the relation on the left we're going to map out the domain and arrange so for the domain we have the values negative 2 1 and 3. for the range we have the y values negative 6 0 and 4. now negative two corresponds to zero one corresponds to four three corresponds to negative six so for every input value on the domain side there's one corresponding output value on a range side so this is a relation i mean this relation is a function so the answer is yes for the first relation now let's move on to the second relation so let's create a mapping diagram as well so let's start with the domain the lowest x value is negative two next we have zero and then the last one is three now looking at the y values the lowest one is negative one and then it's going to be one two and five so negative two corresponds to positive one zero corresponds to five three corresponds to two and zero corresponds to negative one so just by seeing the repeat x values that we see here we could tell that this is not going to be a function the two x values have two different y values you can see zero points to negative one and five so the second relation is not a function now for this one what we're going to do is we're going to draw a function table of the relation and then we're going to determine if the relation is a function so in this table we're going to list the input values next to the output values the input values represent the x values the output values represent the y values so the input values it corresponds to the domain and the output values corresponds to the range so the lowest input value that we have is negative three next is one and then we have another one and then after that is it's three and five now for this function table i'm going to write the input value twice because that's what we have here when writing out the domain and the range for repeat values we would write repeat values once now negative three corresponds to two for the table these numbers need to match so i'm not going to list the output values in ascendant order now for this one we could use either one so i'm going to use 1 2 for the next one and then 1 4. now when x is 3 y is 7 and when x is 5 y is negative 4. so that's the function table and because we have two identical x values that correspond to two different y values we know that this relation is not a function so that's it for this problem when you have a graph the best way to determine if the graph represents a function is to use the vertical line test and that's what we're going to do in this problem so any way you draw a vertical line for the first graph notice that the line only touches the graph at one point therefore this is the answer is yes it represents a function for the next one on the right if we draw a vertical line notice at this point or place a line at that location we have three points of intersection between a graph and a vertical line if we can get two or more points on a vertical line then a relation is not a function so we're going to say no now if we put the vertical line here notice that we have five points on that line so this relation is not a function for the next relation it doesn't matter where we put the graph we will only get i mean doesn't matter where we put the vertical line we're only going to get one point if we put it here it's only going to touch the line once so we can't draw a vertical line where it touches two points therefore this relation represents a function for the circle if we put the line here we can get two points of intersection so we're going to say yes i mean no not yes this is no the circle does not represent a function it does not pass the vertical line test it touches the line at two points in order for it to pass the vertical line test the graph must touch the line only at one point as we saw in these two cases so that's how you can use the vertical line test to determine if a relation represented by a graph is a function