Question Video: Finding the Range of a Piecewise Function given Its Graph Mathematics • Second Year of Secondary School

Video Transcript

Determine the range of the function 𝑓 of 𝑥 is equal to 𝑥 if 𝑥 is greater than or equal to negative two and 𝑥 is less than four and 𝑓 of 𝑥 is equal to eight minus 𝑥 if 𝑥 is greater than or equal to four and 𝑥 is less than or equal to eight.

In this question, we’re given a function 𝑓 of 𝑥, and we can see that this function 𝑓 of 𝑥 is defined piecewise. We need to use this and the given graph of our function 𝑓 of 𝑥 to determine the range of this function. To do this, let’s start by recalling exactly what we mean by the range of a function. We recall that the range of a function means the set of all possible outputs of that function. And there’s something worth highlighting about this definition. The set of possible outputs of a function are going to depend on the inputs we’re allowed for that function. Another way of saying this is the range of a function is dependent on the domain of that function. So we always need to keep this in mind whenever we’re asked to find the range of a function.

To find the range of our function, we actually have two options. Because we’re given a piecewise definition of 𝑓 of 𝑥, we could use this to find the range of our function. However, we can also use the given graph to find this. Either of these will work. However, usually, we have to use both of these together to be sure of our answer.

Let’s start by using just the graph. The range of our function is going to be the set of possible outputs of our function. Remember, the inputs of our function are the values of 𝑥 and the outputs are the corresponding 𝑦-coordinates. For example, we can see from the sketch when 𝑥 is equal to negative two, our function also outputs negative two. And we can also see from our sketch this is the lowest possible output of our function. It doesn’t go any lower.

And this is a very useful piece of information to notice because remember the range is the possible outputs of our function. And we now know that we can’t have a possible output less than negative two. And we can do something very similar to the highest point of our curve. We can see this will be when we input a value of 𝑥 is equal to four. And we can see from our graph that this outputs a value of four. Once again, this is the largest output of our function. So the range of our function can’t include values bigger than four.

And of course, just by looking at our curve, it appears that any possible output between these two values does have an input. For example, if we wanted to check that two was in the range of our function, we would see that the input of two gives us an output of two. Therefore, it appears the possible outputs of our function is any value between negative two and four. We’ve represented this is as the closed interval from negative two to four.

However, we do need to be careful because remember graphs are not always entirely accurate. We can only get approximate values. This means we’re going to need to use the piecewise definition of our function 𝑓 of 𝑥. So let’s start with the first piece of information we notice. We want to know what 𝑓 evaluated at negative two is. We’re going to do this by using the piecewise definition of 𝑓 of 𝑥.

We can see when 𝑥 is equal to negative two, 𝑓 of 𝑥 is equal to 𝑥. So 𝑓 evaluated at negative two is just equal to negative two. And this is exactly what we had in our graph. And in fact, we can notice something very interesting about the piecewise definition of 𝑓 of 𝑥. Whenever our input value of 𝑥 is greater than or equal to negative two and less than four, our function just outputs this value of 𝑥. Therefore, if we input any value of 𝑥 greater than or equal to negative two and less than four into our function 𝑓 of 𝑥, then it just outputs itself. So this entire set has to be possible outputs for our function 𝑓.

But we’re still not done yet. We still want to know what happens for the rest of our input values of 𝑥. And there’s a few different ways we could do this. One way to do this is to notice that eight minus 𝑥 is a linear function. In fact, it has negative slope. And because this has negative slope, as our input values of 𝑥 get bigger, the outputs are going to get smaller. Therefore, when 𝑥 is equal to four, eight minus 𝑥 is going to have its biggest possible output.

So let’s find out what 𝑓 evaluated at four is equal to. We use our piecewise definition of 𝑓 of 𝑥. It’s equal to eight minus four. And of course, eight minus four is just equal to four. Therefore, because 𝑓 evaluated at four is equal to four, we can conclude that four is a possible output of our function 𝑓. And we might want to check the rest of the input values of 𝑥. However, we don’t need to because we can see in our graph these won’t give us any new outputs of our function.

Therefore, because the range is the set of all possible outputs of our function 𝑓, we need to combine these two sets. We can combine all of these possible outputs for our function 𝑓 to just be the closed interval from negative two to four, which is our final answer. Therefore, by using both the graph of our function and the piecewise definition of the function 𝑓 of 𝑥 is equal to 𝑥 when 𝑥 is greater than or equal to negative two and 𝑥 is less than four and 𝑓 of 𝑥 is equal to 𝑥 minus eight if 𝑥 is greater than or equal to four and 𝑥 is less than or equal to eight to show that its range is the closed interval from negative two to four.