### Video Transcript

If ๐ is a map from the set containing negative four, negative one, four, and negative two to the closed interval from six to 25 and ๐ of ๐ฅ is equal to ๐ฅ squared plus five, find the range of ๐.

In this question, weโre given a function ๐ of ๐ฅ, and weโre given an explicit definition of our function. Our function ๐ of ๐ฅ is a quadratic. Given the input ๐ฅ, we get ๐ฅ squared plus five as our output. The question wants us to determine the range of our function ๐. And to do this, weโre going to first need to recall exactly what we mean by the range of a function. The range of a function is the set of all possible outputs of that function. And of course, thereโs one very important piece of information to remember about this. Of course, the outputs of our function are going to depend on the inputs of our function.

And another way of saying this is the range of our function is always going to depend on the domain of our function. In other words, we canโt look at the outputs of a function until we know all the inputs of our function. So the first thing weโre going to want to do is find the domain of our function ๐ or all of the possible inputs of ๐. And this is given to us in the question ๐ is a function from the set containing negative four, negative one, four, negative two to the closed interval from six to 25. So it takes inputs in the set. So this set is the domain of our function ๐; itโs all the inputs weโre allowed.

We might then be tempted to say that the closed interval from six to 25 is the range. However, this would be incorrect. The only thing we can guarantee here is the range will be a subset of the closed interval from six to 25, because the only thing this notation is telling us is ๐ takes the inputs and maps them somewhere into this set. We donโt know how or where theyโre going to be mapped. This is why we need the definition of our function ๐ of ๐ฅ. Weโre now ready to start finding the range of our function ๐. Remember, this will be all of the possible outputs of ๐ given its domain. And the domain only has four entries, so we can do this by just finding ๐ evaluated at each of these entries.

Letโs start with ๐ evaluated at negative four which is the first input given to us in the question. We substitute ๐ฅ is equal to negative four into our definition of ๐ of ๐ฅ. We get negative four all squared plus five. And of course, if we evaluate, this is equal to 21. Therefore, because ๐ evaluated at negative four is equal to 21, we can conclude that 21 must be in the range of ๐ since itโs a possible output of our function. We can do exactly the same with our next input of negative one. ๐ evaluated at negative one is negative one all squared plus five, which we can calculate is equal to six. And just like before, we can now conclude that six must be in the range of our function ๐.

We can do exactly the same for our next input of ๐ฅ is equal to four. ๐ evaluated at four is equal to four squared plus five. And if we were to calculate this expression, we would see that itโs equal to 21. However, this doesnโt actually give us any new pieces of information because we already knew that 21 was in the range of our function ๐. And when weโre finding the range of a function, weโre only interested in the possible outputs of our function. It doesnโt matter if thereโs only one input which gives us output or thereโs multiple inputs which gives us output. All we really needed to know was that at least one input gave this output.

So letโs move on to our final input of ๐ฅ is equal to negative two. We substitute negative two into our definition of ๐. And we get negative two all squared plus five. And we can calculate this; itโs equal to nine. And once again, because thereโs an input which gives us an output of nine, we can conclude that nine must be in the range of our function ๐. But now weโve checked all of the possible outputs of our function ๐ because weโve tried every input in our function ๐. Therefore, these three values are the only possible outputs of our function.

So now we can just give our answer. And remember, the range of a function is a set, so we should give our answer in set notation. And itโs important to remember in a set, it doesnโt matter which order we give our entries. It will be the same set regardless. So weโll give our answer as the set containing nine, 21, and six. However, you can give these numbers in any order you want.

Therefore, we were able to show if ๐ is a function from the set containing negative four, negative one, four, negative two to the closed interval from six to 25 and ๐ evaluated at ๐ฅ is equal to ๐ฅ squared plus five, then the range of our function ๐ will be the set containing nine, 21, and six.