Question Video: Determining the Domain of a Piecewise-Defined Function Mathematics

Video Transcript

Find the domain of the real function 𝑓 of 𝑥 is equal to nine 𝑥 if 𝑥 is in the closed interval from eight to 12, 𝑓 of 𝑥 is equal to 108 if 𝑥 is in the open interval from 12 to 18, and 𝑓 of 𝑥 is equal to 𝑥 plus 90 if 𝑥 is in the left-closed, right-open interval from 18 to 23.

In this question, we’re given a piecewise-defined function 𝑓 of 𝑥 and we’re told that this is a real function. We need to use this to determine the domain of this function. Let’s start by recalling what we mean by the domain of a function. The domain of any function is the set of all input values for that function. So we need to find all of the values of 𝑥 which we can input into our function.

Next, we’re told that our function 𝑓 of 𝑥 is a real function. And this means both the domain and range of our function are subsets of the real numbers. However, we can actually determine both of these pieces of information from the piecewise definition of 𝑓 of 𝑥 we’re given. So we want to determine the domain of our function. Since 𝑓 of 𝑥 is a piecewise-defined function, its input values of 𝑥 are given over separate subdomains. For example, if we input 𝑥 is equal to eight into our function, then 𝑓 of 𝑥 is equal to nine times 𝑥. So 𝑓 evaluated at eight is nine times eight.

We can only input values of 𝑥 in the subdomains of our function. So the domain of our function 𝑓 of 𝑥 will be any value of 𝑥 in any of the three subdomains. And for a value of 𝑥 to be in any of the subdomains, it will be in the union of the three sets. In fact, this is true for any piecewise-defined function. The domain of any piecewise-defined function is the union of its subdomains.

There’s a few different ways of finding the union of these sets. One way is to do this graphically. First, the closed interval from eight to 12 includes all values between eight and 12 inclusive. Next, the open interval from 12 to 18 includes all values between 12 and 18 excluding the endpoints. Finally, the left-closed, right-open interval from 18 to 23 includes all values between 18 and 23 and includes 18 but excludes 23. Combining all of these three intervals together, we include all values between eight and 23, where we include eight but we exclude 23. And this is the left-closed, right-open interval from eight to 23, which is our final answer.

Therefore, we were able to show the domain of 𝑓 of 𝑥 is equal to nine 𝑥 if 𝑥 is in the closed interval from eight to 12, 𝑓 of 𝑥 is equal to 108 if 𝑥 is in the open interval from 12 to 18, 𝑓 of 𝑥 is equal to 𝑥 plus 90 if 𝑥 is in the left-closed, right-open interval from 18 to 23 is the left-closed, right-open interval from eight to 23.