Video: Finding the Range of a Function Represented by a Cartesian Diagram

The figure below shows the graph of a function 𝑓. What is the range of the function?

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Video Transcript

The figure below shows the graph of a function 𝑓. What is the range of the function?

In this question, we’re given the graph of a function 𝑓 and we’re asked to determine the range of this function. So to do this, let’s start by recalling what we mean by the range of a function. We recall that when we say “the range of a function,” what we mean is the set of all possible outputs of this function. And there is something worth highlighting about this definition. The set of all possible outputs of a function is going to depend on what inputs were allowed. In other words, the range of a function depends on the domain of this function.

So when the question asks us to find the range of this function, we want to determine all of the possible outputs that our function can give us. So we’re going to want to do this from our graph. And remember, in a graph, the 𝑥-values are the inputs and the 𝑦-values are the outputs of our function. And even before we start looking for the range of our function, we can notice a few interesting things about this graph.

The first thing we can notice is our function is only four defined points. We are used to seeing functions which are curves or lines. However, in this case, our function is only four dots. And each of these points is going to represent an input value of 𝑥 and an output value of 𝑦. Therefore, because there’s only four points, our domain is going to have four input values of 𝑥. And as we’ll see, this doesn’t necessarily mean the range is going to have four values.

The next thing we can notice is our axes are not meeting at the origin. This won’t change anything; however, it is useful to notice this piece of information.

We’re now ready to use our graph to find the range of our function 𝑓. To do this, we’ll find the possible inputs of our function and then use our graph to find the outputs of our function. Let’s start with the leftmost point on our diagram. We can see that the 𝑥-value of this point is one. Therefore, we’re allowed to input the value of one into our function. And remember, the corresponding 𝑦-coordinate of this point will tell us the output of our function when 𝑥 is equal to one. And we can see this is negative two. Therefore, 𝑓 evaluated at one is equal to negative two. So we’ve shown that negative two is in the range of our function because it’s a possible output.

Let’s now move on to the second point. We can see that this has an 𝑥-coordinate of two. Therefore, two is a possible input of our function. Once again, we can use the 𝑦-coordinate of this point to evaluate 𝑓 at two. And the 𝑦-coordinate of this point is negative three. Therefore, we know from the graph of this function 𝑓 evaluated at two must be equal to negative three. And of course, this tells us that negative three is also in the range of our function because it’s a possible output of our function.

We can then do exactly the same thing for our third point. We can see its 𝑥-coordinate is equal to three. So three is a possible input of our function. And remember, this also means that three is in the domain of our function. And the 𝑦-coordinate of this point will tell us the output of our function when we input a value of three. And we can see this 𝑦-coordinate is zero. Therefore, from our graph, we must have that 𝑓 evaluated at three is equal to zero. And once again, this also tells us that zero is in the range of our function.

Now we move on to our fourth and final point. We can see its 𝑥-coordinate is equal to four. So four is a possible input of our function. In other words, four is in the domain of our function. And we want to see the output of our function when 𝑥 is equal to four. So we need to find the 𝑦-coordinate of this point. We can see it’s equal to negative three. Therefore, from our graph, we must have that 𝑓 evaluated at four is equal to negative three. And this tells us that negative three is in the range of our function. However, we already knew that negative three was in the range of our function because 𝑓 evaluated at two is equal to negative three.

And when we’re finding the range of a function, we need to know the set of all possible outputs of our function. We don’t need to know how many input values of 𝑥 give us our outputs. All we’re interested in is at least one input value of 𝑥 gives us this output. So we’ve only found three output values of our function. And it’s worth pointing out here we don’t need to check any more input values of 𝑥 because we only need to check the input values of 𝑥 where our graph lies and we can see that the graph is only four points. So we only need to check these four points.

So these three values are the only possible outputs of our function. They’re going to make up the range of our function. And remember the range of a function is a set. So we should give our answer in set notation. We’ll write the range of our function as the set containing negative three, negative two, and zero. And of course, because this is a set, we can write the entries of our set in any order. It won’t change our answer. It’s all personal preference. Therefore, given a graph of the function 𝑓, we were able to determine the range of our function. It was the set containing negative three, negative two, and zero.

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