Lesson Video: Domain and Range from Function Graphs Mathematics

Video Transcript

In this video, we will learn how to identify the domain and range of functions from their graphs. First of all, we’ll think about the definition of domain and range. If we let this machine represent some function machine, the domain will be the values that we begin with. The domain is the complete set of possible values, and these values are independent. It is the independent variable value. And on a standard coordinate grid, these are going to be the 𝑥-values. The 𝑥-axis represents the independent variables. And the range is the complete set of all possible resulting values. It is the dependent variable. And on a standard coordinate grid, this is the 𝑦-value. The 𝑦-values are the output values of this function. The 𝑥-values are the inputs, and the 𝑦-values will be the output.

To explore this further, we’ll start looking at some graphs and some example problems.

The domain of the function 𝑓 of 𝑥 is blank.

The function 𝑓 of 𝑥 here is represented by these five points. First, we remember that the domain is the set of all possible 𝑥-values for a function. And then we recognize that on a coordinate grid, the 𝑥-axis is the horizontal axis, which means the 𝑥-values of this function will be found by looking at where these points fall horizontally. All the way to the left, we have a point at negative seven. To the right, we have a point at negative six, followed by negative five, negative four, and negative three.

It’s important to notice that these points are not connected with the line. Because of that, we know that this is not a continuous function and that the domain is then just going to be a list of the possible 𝑥-values. In set notation, it would look like this: negative seven, negative six, negative five, negative four, and negative three.

If we wanted to, we could consider the range as well. The range will be the possible 𝑦-values of this function. And that will be how far the points are located up or down, where they fall on the vertical axis. For this function, we have 𝑦-values of one, two, three, four, and five. And set notation for the range would look like this: one, two, three, four, five.

As this question was only asking for the domain, the domain of 𝑓 of 𝑥 here is the set negative seven, negative six, negative five, negative four, and negative three.

Let’s look at another example.

Determine the domain and range of the function 𝑓 of 𝑥 equals negative four.

In the image, we’ve been given the graph of the function 𝑓 of 𝑥 equals negative four. In order to calculate the domain and range, we remember that the domain is represented by the 𝑥-values and the range is represented by the 𝑦-values on the graph. We also remember that the domain is the independent variable. It’s the variable we plug in to our function. We want to know what is the set of values that 𝑥 can be.

Now, on this graph, it might look like 𝑥 only goes from negative four to positive four. However, we recognize that this is a function that continues in both directions. To the right, 𝑥 would continue out to positive ∞ and to the left negative ∞. So how should we write this as a domain?

We could use this symbol that looks a little bit like an R. This symbol represents all real numbers. The domain for 𝑥 can be any real number.

What about the range? The range is a little bit different here. The range will be the 𝑦-values, that is, the distance up or down from zero. For every 𝑥-value in this function, 𝑦 is always negative four. 𝑦 does not change. And that means the only outcome, the only output of this function, is negative four. The range is the set of negative four. And so we can say for the function 𝑓 of 𝑥 equals negative four, the domain is all real numbers and the range is the set negative four.

In our next example, we’re given the graph of a cubic function and we’ll need to find its domain and range.

Find the domain and range of the function 𝑓 of 𝑥 equals 𝑥 minus one cubed in all reals.

We’ve already been given the graph of this function, 𝑥 minus one cubed. So now we just need to think about what the domain and range are. When we have a graph, the domain is represented by the set of possible 𝑥-values and the range is the set of all possible 𝑦-values. It’s important to know that when we have this type of graph, we know that they continue in both directions. While we’re only seeing a bit of this function, from 𝑥 negative two to 𝑥 positive three, we know that it continues in both directions. The same thing is true for the 𝑦-values. We’re only seeing 𝑦-values up to positive 10 and down to negative 10.

However, this function continues outside of this window on our graph. In this case, we have no limits on our domain or range. The domain can be all real numbers, and the range can be all real numbers. It’s also possible that we might want to write this in interval notation instead of in set notation. The interval of the domain would be written as negative ∞ to ∞. And in this case, the same thing will be true for the range interval, all real numbers or values from negative ∞ to positive ∞.

With the interval notation here, it’s important to note that we use the round brackets when we are not including what is on the end. So what these say is that we want to go up to ∞ but not including ∞.

For our next example, we’ll look at determining the domain and the range of a piecewise function.

Determine the domain of the following function.

We know that the domain of this function will be the set of all possible 𝑥-values. And on a coordinate grid, that is the 𝑥-axis, the horizontal axis. We see designated values from negative seven all the way to positive seven. However, we should know that the arrows on either side of this graph indicate that this function continues. On the left, we would say that the graph could continue to negative ∞ and on the right to positive ∞.

However, let’s think carefully about what’s happening at zero. When 𝑥 equals zero, does this function have a result? We know that it does because the point is colored in at zero, four. Zero, four is a result, but zero, negative four is not filled in and is therefore not a result of this function. Since we do have a result at zero, we can confirm that there’s a domain of all real numbers.

This question hasn’t asked us for a range. But if we wanted to add the range, that would be the output values, the set of possible 𝑦-values. And we see that there are two possible values: one value at four and one value at negative four. In set notation, we could write that the range is therefore negative four and four. As the question has only asked us to identify the domain, we can simply say that the domain is all reals.

In our final example, we’ll look at a graph where there are limits to the domain and the range.

Find the domain and range of the function 𝑓 of 𝑥 is equal to negative one over 𝑥 minus five.

We’ve already been given a graph of this function. And we can use the graph to identify both the domain and the range of the function. The domain will be the set of all possible 𝑥-values. And on this graph, we can use the 𝑥-axis to identify that. And the range will be the set of all possible 𝑦-values. We’ll use the 𝑦-axis to identify that.

But before we do that, let’s carefully consider the behavior of the function in the graph we’re looking at. We see that it kind of has two pieces: one above the 𝑥-axis and one below the 𝑥-axis. And then we have this dotted line. When we have a dotted line like this on the graph, it represents an asymptote of the function. An asymptote is a line that a curve approaches as it heads towards ∞. The curve will never cross the asymptote. And this asymptote is located at 𝑥 equal to five. And that means we can say, for sure, that the domain does not include the value 𝑥 equals five.

But if we look at the rest of the function, we can see that there are 𝑥-values extending in the left and right direction. And so 𝑥 can be anything except for positive five, which means the domain is all reals minus the set five. Now, if we’re thinking about the range, we’re thinking about the vertical behavior of our graph. And again, we notice that there is one piece of this graph above the 𝑥-axis and one piece below it. Even though they didn’t add a dotted line, the 𝑥-axis represents another asymptote of this function. The 𝑦-value of this function is getting closer and closer to zero, but it is never crossing zero. And that is true on both the left- and the right-hand side of this function. This means that the 𝑦-values can be anything except for zero.

And so, in a similar format, we say that the range will be all reals minus the set zero. The set of five in the domain and the set of zero in the range represents the vertical and horizontal asymptotes of this function, and we’d correctly label the domain and range.

Before we finish, let’s review some key points from this video. The domain of a function is the complete set of possible values of the independent variable. The range of a function is the complete set of possible resulting values. Given the graph of a function, the domain is all possible 𝑥-values and the range is all possible 𝑦-values.