Introduction to Functions - Online Tutor, Practice Problems & Exam Prep

Relations and Functions

Hey, everyone. So, up to this point, we've spent a lot of time talking about graphs. And in this video, we're going to see if we can apply this concept of graphs to this new topic of relations and functions. Now, this topic is often considered confusing when students initially encounter it, but throughout this video, we're going to be going over a lot of different scenarios and examples to see if we can really clear up some of this confusion around this subject. So let's get right into this.

Relations are a connection between X and Y values and graphically they are represented as ordered pairs. Now, functions are a special kind of relation where each input has at most one output, and it's important to note that all functions are relations, but not all relations are functions. So, to understand this a little bit better, let's take a look at this example that we have down here. We have these two graphs of relations and we want to see if we can determine whether these are also examples of functions. We'll start with this graph on the left, and what I'm going to do is write out all of the inputs which correspond to all of the x values.

So, I'll do this in ascending order. So first, I see that we have an x value of negative 2. I also see that we have an x value of positive 1, and even though it shows up twice here it's perfectly okay to only write it once in this bubble down here. Now lastly, I see that we have an x value of 3. So these are all of the inputs that we have.

Now, the outputs are going to correspond to the Y values, and I'll list all of these out as well. So I see that we have positive 2, I see that we have 4, we also have 1, and then we have negative 2. So these are all of the inputs and outputs. Now, looking at these points I see that negative 2 is related to positive 2, I see that positive one is related to 4, and I also see that one is related to 1. Now, lastly, I see that we have this point which says 3 is related to negative 2.

Now, based off this relation that we see, can we conclude whether or not it's a function? Well, recall that we set up here for a relation to be a function each input can have at most one output. And if I look at each of our inputs, I can see that there is an input that has more than one output. And as soon as this happens, you could automatically conclude that this is not an example of a function. But let's take a look at this other example for this graph on the right.

So what I'm going to do with this graph is I'm going to list out all of the inputs in ascending order like we did before. So this will be all the X values, so I see that we have negative 4, I see that we have negative 2, I see that we have 1, and then we have 3. Now what I'm also going to do is list out all of the outputs, which correspond to all the Y values. So I see that we have positive 2, I see that we have negative 1, and I also see that we have positive 2 up here, but since we already wrote positive 2 once we don't have to write it again. So we have positive 2, and then we have positive 4.

So these are all of the outputs. Now, looking at how these are related, I can see that negative 4 is related to negative or to excuse me to positive 2. I see that negative 2 is related to negative 1, and I see that positive 1 is related to positive 2, and then I see that we have that positive 3 is related to positive 4. Now, given this information, can we conclude whether or not this is a function? Well, we need to see if any of the inputs have more than 1 output, and if I look at this each of these inputs only go to one output, and because of this, we would say this is an example of a function.

So this is how you can tell whether or not a relation is a function, but you may have noticed this process was a bit tedious having to write out all the inputs and outputs like this. Well, you may be happy to know there is a shortcut to solving these problems, and the shortcut is called the vertical line test. This states that if you can draw any vertical line that passes through more than one point on your graph, then the graph is not going to be a function. So let's try this vertical line test on the two graphs we had up here. I'll take vertical lines and I'll draw them through every point that I see on the graph, and let me draw this vertical line a bit better.

So if I draw these vertical lines, I notice that there is a place where the vertical line passes through more than one point. If this ever happens, then it's not a function. But let's try the vertical line test on this other graph. If I draw vertical lines through each of the points that I see, I notice that no matter where I draw a vertical line, I'm only ever going to pass through one point at most. And because of this, we would say that this is an example of a function.

Now let's take a look at a couple more examples, because you could also use the vertical line tests on graphs like this. If I tried drawing some vertical lines for this graph on the left, I notice that we will have some vertical lines that pass through more than one point, and because of this, we can conclude that this is not an example of a function. But if I try the same vertical line test on the other graph that we have over here, notice that no matter where I draw a vertical line, we're only ever going to pass through one point at most, and because of this, we can conclude that this graph is an example of a function. So that's the basic idea of relations and functions, hopefully, this helped you out, and let me know if you have any questions.

State the inputs and outputs of the following relation. Is it a function? {$\left(-3,5\right),\left(0,2\right),\left(3,5\right)$}

• A.
• B.
• C.
• D.

State the inputs and outputs of the following relation. Is it a function? {$\left(2,5\right),\left(0,2\right),\left(2,9\right)$}

• A.
• B.
• C.
• D.

Relations & Functions Example 1

So let's see if we can solve this problem. In this problem, we have 3 different graphs. We're asked to determine which of the graphs are functions, and we can select all that apply. Whenever you want to figure out whether or not a graph is a function, you can use the vertical line test. If you can draw a vertical line somewhere on the graph that crosses through more than one point, it is not a function.

However, if the vertical line you draw can only cross through one point at most anywhere you draw it, it is a function. So let's see if we can figure this out. Now for this first graph, we'll try the vertical line test. If I draw a vertical line here, we only cross through one point. If I try a vertical line over here, we only cross through one point.

If I try a vertical line through the center or close to the center, we only cross through one point. And it turns out even though this graph expands in both directions, anywhere you would draw a vertical line would only ever cross through one point. So we can see that this first graph, graph a, is an example of a function. But let's try graph b over here, and I'll go ahead and draw a vertical line. Notice the vertical line we drew crosses through more than one point.

Since we see that we can draw a vertical line somewhere that crosses through more than one point, that means this is not an example of a function. But now let's try the vertical line test on this graph over here. Well, if I draw a vertical line there, we only cross through one point. If I draw a vertical line here, we only cross through one point, and anywhere I would draw this vertical line would only ever cross through one point. So we would say that graph c is an example of a function.

So to list all of the graphs that are functions, we would say graph a and graph c are functions. That's the answer to this problem.

Finding the Domain and Range of a Graph

Hi everyone, and welcome back. Up to this point, we have spent a lot of time talking about functions. In this video, we're going to be taking a look at how we can find the domain and range of a function based on its graph. Now, when encountering the task of finding the domain and range of functions, it can often seem a bit tricky. But in this video, we are going to go over some analogies and examples that will hopefully make this topic a lot clearer.

So let's get into this. When finding the domain of a graph, you're looking for the allowed x values that you can have. And when finding the range, you're looking for the allowed y values. Now, there's a little trick that you can use, and that is known as the squish strategy. What you can do if you have the domain of a graph is you can take your graph and squish it down to the x axis.

So, if you squish things down to the x axis, that will tell you the domain. Let's say we have this graph down here. Notice how we took this curve and we literally squished it down to the x axis so only some sort of line or set of lines remains. This tells you the domain. Now, when finding the range, you can take your graph and squish it down to the y axis.

Notice for our graph, we've squished it down to the y, and this tells you the range. There are a couple of different ways we can write the domain and range of a graph. One common method is to use interval notation. Interval notation represents the domain and range using brackets and parentheses. You can actually see this in the example we have above.

Notice that it looks like our domain goes from a value of negative 4 to positive 5 on the x axis. This is what we stated the domain was. And for our range, it looks like we go from about negative one to positive 2 on the y axis. That's using interval notation. Another notation you can use is called set builder notation.

Set builder notation uses inequality symbols as opposed to the brackets and parentheses. Something to keep in mind whenever using either of these notations is that whenever you see these brackets or inequality symbols with the bar underneath, it means that you want to include whatever value you're looking at. For example, if we have this graph, the values you want to include are either closed dots or solid lines or curves. These are values that you always want to include in your domain and range.

Whenever you see these parentheses or inequality symbols without the bar underneath, this means that you do not want to include those values. Values that you do not want to include are situations where you see a hole in your graph. Anytime you see this hole or this open circle, that means you do not want to include it, whereas anytime you see a solid dot or a solid line or curve, you do want to include it. Given all this information, let's see if we can solve an example. We're asked to determine the domain and range of the graph below to express our answer using interval notation.

First, let's find the domain of this graph. Recall that what we want to do is take our graph and squish it down to the x axis. If we take this graph and we imagine squishing every single point down to the x axis, we're going to get a graph that looks something like this. We'll have a line that goes from negative 4 all the way to an x value of 0 and then we'll have another line that goes from positive one all the way to 4.

Notice how there's an open circle here, so we have to include the open circle there as well. And then everywhere else, we have a closed circle because these circles were all closed. Looking at this squished graph, our domain is going to go from negative 4 to 0 on our x axis and we need to include both these values since we have 2 closed circles. Another one of these lines is going to go from positive one because this is an x value of 1 right here to 4. We're going to include this positive one since we have a closed circle there, but our open circle is going to cause us to draw a parenthesis here because we do not want to include that value.

Notice how we have multiple intervals here. Whenever you have multiple intervals or a jump in your graph, you need to use something called a union symbol, which looks like this. Since we have multiple intervals, we need to put a union symbol between these to join the 2 intervals together. This right here is the domain of our graph.

Now let's see how we can find the range. And recall when finding the range, we need to take our graph and squish it down to the y axis. If I take this other graph we have drawn over here and I squish all these values down to the y, what we're first going to end up with is a line that goes from 1 to 3 on the y axis and then we're going to end up with another line that goes from negative 3 all the way to negative 1 and we'll have an open circle at negative 1 since we have the open circle right there. So our range is going to go from negative 3 to negative one and keep in mind we'll include the negative 3 but we do not include the negative one since we have an open circle, and our range will also go from positive one to 3. And we need to include both of these values since we have closed dots or, I should say, we have a solid line or curve, but still, you want to include that value.

We'll put the union symbol, and then this is the range of our graph. So that is how you can find the domain and range of a function based on its graph. Let me know if you have any questions, and thanks for watching.

Find the domain and range of the following graph (write your answer using interval notation).

• A. Dom: $\left[-5,5\right]$ , Ran: $\left\lbrack-4,4\right\rbrack$
• B. Dom: $\left[-2,2\right]$ , Ran: $\left[-3,3\right]$
• C. Dom: $\left[-4,4\right]$ , Ran: $\left[-5,5\right]$
• D. Dom: $\left(-5,5\right)$ , Ran: $\left(-4,4\right)$