Question Video: Finding the Range of Quadratic Functions Mathematics

If ๐‘“ : {โˆ’4, โˆ’1, 4, โˆ’2} โŸถ [6, 25] and ๐‘“(๐‘ฅ) = ๐‘ฅยฒ + 5, find the range of ๐‘“.

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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 ๐‘ฅ is equal to ๐‘ฅ squared plus five, find the range of ๐‘“.

In this question, weโ€™re given a function ๐‘“ of ๐‘ฅ, and weโ€™re given an explicit definition of our function. Our function ๐‘“ of ๐‘ฅ is a quadratic. Given the input ๐‘ฅ, we get ๐‘ฅ squared plus five as our output. The question wants us to determine the range of our function ๐‘“. And to do this, weโ€™re 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, thereโ€™s 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 canโ€™t look at the outputs of a function until we know all the inputs of our function. So the first thing weโ€™re 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‘“; itโ€™s all the inputs weโ€™re 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 donโ€™t know how or where theyโ€™re going to be mapped. This is why we need the definition of our function ๐‘“ of ๐‘ฅ. Weโ€™re 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.

Letโ€™s start with ๐‘“ evaluated at negative four which is the first input given to us in the question. We substitute ๐‘ฅ is equal to negative four into our definition of ๐‘“ of ๐‘ฅ. 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 itโ€™s 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 ๐‘ฅ 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 itโ€™s equal to 21. However, this doesnโ€™t actually give us any new pieces of information because we already knew that 21 was in the range of our function ๐‘“. And when weโ€™re finding the range of a function, weโ€™re only interested in the possible outputs of our function. It doesnโ€™t matter if thereโ€™s only one input which gives us output or thereโ€™s multiple inputs which gives us output. All we really needed to know was that at least one input gave this output.

So letโ€™s move on to our final input of ๐‘ฅ 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; itโ€™s equal to nine. And once again, because thereโ€™s 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 weโ€™ve checked all of the possible outputs of our function ๐‘“ because weโ€™ve 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 itโ€™s important to remember in a set, it doesnโ€™t matter which order we give our entries. It will be the same set regardless. So weโ€™ll 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 ๐‘ฅ is equal to ๐‘ฅ squared plus five, then the range of our function ๐‘“ will be the set containing nine, 21, and six.

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