Lesson Video: Domain and Range of a Piecewise Function Mathematics

Video Transcript

In this video, we will learn how to find the domain and the range of a piecewise-defined function. Let’s begin by recalling what we mean by these terms, domain and range. The domain of a function is the set of all input values to the function. We can think of this as the set of all values on which the function acts. The range of a function is the set of all possible outputs of the function given its domain. We can think of this as the set of all values the function produces.

On a graph of the function, the domain and range correspond to the sections of the 𝑥- and 𝑦-axes for which the graph is plotted. For example, if we consider the function 𝑓 of 𝑥 equals 𝑥 squared, we can see that every single 𝑥-value in the real numbers is inputted to the function. And so the domain is the entire set of real numbers. But if we consider the range of this function, we can see that there are certain 𝑦-values, the 𝑦-values below the 𝑥-axis, which are not reached by the graph. Only 𝑦-values above or on the 𝑥-axis are reached by the graph. So we would say that the range of this function is the set of all values of 𝑦 or 𝑓 of 𝑥 which are greater than or equal to zero.

Now, in this video, we’re looking specifically at the domain and range of piecewise functions. So let’s also recall what we mean by that. A piecewise function is one which is defined differently over different subsections of its domain. For example, we could have the function 𝑓 of 𝑥, which is defined as being equal to one when 𝑥 is less than zero and equal to two when 𝑥 is greater than or equal to zero. This is a very simple piecewise function. It has only two subdomains: 𝑥 is less than zero and 𝑥 is greater than or equal to zero. And the subfunctions themselves are also very straightforward. They’re just constants.

There is, however, no limit to the number of subdomains a piecewise function can have. And the subfunctions themselves could also be much more complex. They could be polynomials or more complicated even than that. Formally then, we can say that a piecewise function is a function that consists of multiple subfunctions, with each subfunction defined over a subset of the main function’s domain, which we call the subdomain.

Let’s look at how we might find the domain and range of a piecewise function from its graph first of all. So here we have a piecewise function. We aren’t given the definition of the function, although we could probably work it out from the graph if we needed to. To find the domain of this function first, we need to look at all the 𝑥-values for which the function has been plotted. We can see that the graph has two sections: the horizontal section here and then the diagonal section here. So we want to consider the 𝑥-values for each of these, which will give the subdomains of the function.

The open circle here indicates that the 𝑥-value at this point, which is negative two, is not included in this subdomain because this point is not on this part of the graph. We can see though that this blue line has an arrow on the end pointing towards the negative 𝑥-values, which indicates that the line continues infinitely in this direction. The subdomain for this section of the graph then is all 𝑥-values strictly less than negative two.

Let’s now consider the other section of the graph. This starts at an 𝑥-value of negative two. And this time, the closed or solid circle indicates that this point is on the graph for this section. And so this subdomain does include the value negative two. At the other end, the line extends to an 𝑥-value of two. But the open dot here indicates that this point and, hence, this value of 𝑥 is not included. So we have all values of 𝑥 greater than or equal to negative two but strictly less than positive two.

Now, here’s an important key point. The domain of a piecewise function is the union of all of its subdomains. So any 𝑥-value that’s in one of the subdomains is also in the domain of the overall function. We can use interval notation if we wish to say that the domain of this function is the union of the open interval from negative ∞ to negative two and the left-closed, right-open interval from negative two to positive two. That gives the open interval from negative ∞ to two. Or we can write this using inequalities as 𝑥 is strictly less than two.

So we found the domain of this function, and now let’s consider the range. To do this, we need to consider all the 𝑦-values that are reached by the graph. In a similar way to the domain, the range of a piecewise function is the union of the ranges of each subfunction over its subdomain. Looking at the graph, we see that the first subfunction is constant because the line is horizontal. And the 𝑦-value is always equal to negative five. So this is the range of the first subfunction.

The second subfunction is a continuous straight line, which extends from a 𝑦-value of negative three to a 𝑦-value of positive five, although the open dot here indicates we can have all values up to but not including positive five. So the range of this piecewise-defined function is the union of the value negative five with the left- closed, right-open interval from negative three to positive five.

For more complicated piecewise functions, we can find the domain and range by considering where the vertical and horizontal lines intersect the curve. For example, if we sketch in the vertical line 𝑥 equals one and this does intersect the curve, then this tells us that the 𝑥 or input value of one is in the domain of the function. If we draw a horizontal line at, for example, 𝑦 equals three and this line intersects the curve, then this tells us that this 𝑦-value is in the range of the function. If, however, we were to draw a line at, for example, 𝑦 equals negative four, then this horizontal line does not intersect the curve. And so this tells us that the value of negative four is not in the range of the function.

We need to be able to determine the domain and the range of a piecewise function both graphically and from its definition. In our first example, we’ll practice finding the domain and range of a piecewise function given its graph.

Determine the domain and the range of the function 𝑓 of 𝑥 equals six when 𝑥 is less than zero and negative four when 𝑥 is greater than zero.

We recall first that the domain of a function is the set of all input values to that function. In the case of the piecewise-defined function we have here, the domain will be the union of all of its subdomains. We can determine this either from the definition of the function or from the graph. Looking at the function itself first, we see that 𝑓 of 𝑥 is defined to be equal to six when 𝑥 is less than zero and negative four when 𝑥 is greater than zero. In other words, the function acts on all negative 𝑥-values to produce the output six and acts on all positive 𝑥-values to produce the output negative four. So the first subdomain of the function is 𝑥 is less than zero, and the second subdomain is 𝑥 is greater than zero.

The domain of 𝑓 of 𝑥 is the union of these two subdomains, which we can write in interval notation as the union of the open interval from negative ∞ to zero and the open interval from zero to ∞. This is the entire set of real numbers with only the value zero excluded. So we can write this as the set of real numbers minus the value zero.

Another way to see this is from the graph. To determine if a value of 𝑥 is in the domain of this function, we can draw a vertical line at this 𝑥-value. If this line intersects the graph, then this tells us that this value of 𝑥 is in the domain of the function. The only vertical line we can draw that doesn’t intersect the graph of 𝑦 equals 𝑓 of 𝑥 is the line 𝑥 equals zero. So this is the only value of 𝑥 excluded from the domain.

Next, let’s consider the range, which is the set of all output values produced by the function. From the definition of the function, we see that the function can only take the values six and negative four. These are, therefore, the only outputs of the function, and so they’re the only values in the range. We can also see this from the graph if we draw horizontal lines at different 𝑦-values. The only horizontal lines that intersect the graph are the lines 𝑦 equals six and 𝑦 equals negative four. So the range of 𝑓 of 𝑥 is the set of values negative four, six.

We’ve found then that the domain of this piecewise function is the set of all real numbers minus the value zero and the range is the set containing the values negative four and six. Let’s now consider another example in which we find the range of a different piecewise-defined function.

Find the range of the function 𝑓 of 𝑥 equals 𝑥 plus five for 𝑥 in the closed interval from negative five to negative one and 𝑓 of 𝑥 equals negative 𝑥 plus three for 𝑥 in the left-open, right-closed interval from negative one to three.

We recall that the range of a function is the set of all possible output values of the function given its domain. We have here a piecewise-defined function. It’s defined differently over different subdomains. To find the range of this function though, we can use its graph. And we can consider which horizontal lines will intersect the graph. The minimum value of 𝑦 at which we can draw a horizontal line that intersects the graph is zero. And the maximum value of 𝑦 at which we can draw a horizontal line that intersects the graph is four. Any horizontal line we draw between these two 𝑦-values will also intersect the graph, whereas any horizontal line we draw outside these two 𝑦-values will not intersect the graph. This tells us that the range of this piecewise-defined function, which is the set of all possible output values or 𝑦-values on its graph, is the closed interval from zero to four.

Let’s now consider a different example in which we’ll determine the domain of a piecewise function, but this time we won’t be given its graph.

Determine the domain of the function 𝑓 of 𝑥 equals 𝑥 plus four when 𝑥 is in the closed interval from negative four to eight and 𝑓 of 𝑥 equals seven 𝑥 minus 63 when 𝑥 is in the left-open, right-closed interval from eight to nine.

We recall first that the domain of a function is the set of all input values to that function. And for a piecewise-defined function, as we have here, it is the union of its subdomains. For this function, the subdomains are the closed interval from negative four to eight and the left-open, right-closed interval from eight to nine. So the domain of the function is the union of these two intervals. There’s no gap between these two intervals, and the value of eight is included because the first interval is closed at the upper end. So the domain of this piecewise function, which is the union of its subdomains, is the closed interval from negative four to nine.

Let’s now consider one final example in which we’ll determine both the domain and the range of a slightly more complex piecewise-defined function.

Determine the domain and the range of the function 𝑓 of 𝑥 equals 𝑥 squared minus 36 over 𝑥 minus six if 𝑥 is not equal to six and 𝑓 of 𝑥 equals 12 if 𝑥 is equal to six.

We recall first that the domain of a function is the set of all input values to that function. And for a piecewise-defined function, as we have here, it is the union of its subdomains. To find the union of the subdomains, we’ll start by writing them in terms of sets. Firstly, 𝑥 not equal to six is the same as the set of all real numbers minus the value six. Second, 𝑥 equals six is simply the same as the set containing the value six. So the domain is the union of these two sets: the real numbers minus the value six union the value six, which is just the complete set of real numbers.

Next, let’s consider the range of this function. The range is the set of all possible outputs given the function’s domain. For a piecewise-defined function, this will be the range of the subfunctions over their subdomains. So we can determine the range of this function by considering the range of each subfunction separately.

First, let’s consider the definition of 𝑓 of 𝑥 when 𝑥 is not equal to six. The expression in the numerator is a difference of two squares. And so it can be factored as 𝑥 minus six multiplied by 𝑥 plus six. The shared factor of 𝑥 minus six in the numerator and denominator can be canceled. And this is why it’s important that 𝑥 is not equal to six here, because if it was, then 𝑥 minus six would be equal to zero, and we’d be dividing by zero, which is undefined. But as the function is defined differently when 𝑥 is equal to six, we can be confident we’re not attempting to divide by zero here.

So when 𝑓 of 𝑥 is not equal to six, it simplifies to simply 𝑥 plus six. If we wish, we can then sketch this subfunction. It’s the line 𝑦 equals 𝑥 plus six with the point when 𝑥 is equal to six removed. The range of this subfunction is all of the possible outputs. The only horizontal line we can draw that doesn’t intersect this graph is the line 𝑦 equals 12. So the range of this subfunction is the set of all real numbers minus the value 12.

However, the second subfunction is the constant function 𝑓 of 𝑥 equals 12 on the domain 𝑥 equals six. Since the output is constant, the range of the second subfunction is simply 12. Taking the union of the ranges of these subfunctions gives the set of real numbers minus the value 12 union the value 12, which is simply the set of all real numbers. We’ve taken the value 12 out and then put it back in again.

We can also sketch the second subfunction on the same graph to fully sketch 𝑓 of 𝑥. The second subfunction is only defined when 𝑥 equals six. So it consists of a single point. 𝑓 of six is equal to 12, so we can add the point six, 12 to our sketch. We filled in the point that was previously missing from the line. So we can see that the function 𝑓 of 𝑥 is 𝑥 plus six. We can conclude then that the domain of this piecewise-defined function is the complete set of real numbers and so is the range.

Let’s finish by summarizing some of the key points from this video. First, we saw that the domain of a piecewise-defined function is the union of its subdomains. In the same way, the range of a piecewise-defined function is the union of the ranges of each subfunction over its subdomain. We saw that to find the domain of a function from its graph, we can consider the intersections of the curve with the vertical lines. If a vertical line does intersect the curve, then that 𝑥-value is included in the domain, whereas if a vertical line doesn’t intersect the curve, that 𝑥-value is excluded.

To find the range of a function from its graph, we can consider the intersections of the curve with horizontal lines and apply the same principles to determine whether values of 𝑦 are included or excluded from the range of the function.