Determine the range of the function represented by the figure in 𝑥.
In this question, we’re given a function which is represented by the given figure. We need to use this to determine the range of this function. To do this, we’re first going to need to recall exactly 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 that function. And there is one thing worth highlighting about this definition. The possible outputs of a function are going to depend on the inputs we’re allowed for our function. In other words, the range of our function depends on the domain of our function.
Then let’s try and find the range of our function. We want to find all of the possible outputs for this function. And to do this, we’re going to need to use the figure given to us. In this figure, when we have a value of 𝑎 and a value of 𝑏 connected by an arrow, then 𝑎 represents our input and 𝑏 represents the output of our function. In other words, our value of 𝑎 is being mapped to the value of 𝑏. And since we want to find the range of our function, we want to know all the possible outputs of this function. So we want to know all of the values which have an arrow pointing to them.
So let’s go from left to right. Let’s start with negative five. We can see that negative five is indeed in the range of our function, and this is because negative four maps to negative five. Next, let’s check negative four. Once again, we can see that negative four is indeed in the range of our function. And this time, we can see this is true because negative five maps to negative four. Next, we’ll want to see if negative three is in our range. And by looking at the figure, we can see that negative three maps to give us negative three. So negative three is a possible output of our function, and therefore it must be in the range of our function.
However, if we were to check the value of negative two, we see that there’s no arrows pointing to negative two. This means that no input value will map to our value of negative two. Therefore, it’s not in the range of our function. And finally, we could see we run out of possible values in our figure. So these are all of the possible values of our range. So the range of our function only contains the three values, negative five, negative four, negative three.
But remember, the range of a function is given as a set, so we should write this in set notation. This gives us a set containing negative five, negative four, negative three, which is our final answer. Therefore, we were able to determine the range of the function represented by the figure given to us in the question. The range of this function was the set containing negative five, negative four, and negative three.