Video Transcript
Find the domain of the function 𝑓 of 𝑥 is equal to 𝑥 when 𝑥 is in the left-closed, right-open interval from zero to two and 𝑓 of 𝑥 is equal to two when 𝑥 is in closed interval from two to six and 𝑓 of 𝑥 is equal to eight minus 𝑥 when 𝑥 is in the left-open, right-closed interval from six to eight.
In this question, we’re given a piecewise function 𝑓 of 𝑥, and we’re asked to determine the domain of this function. And we can start by recalling the domain of a function is the set of all input values for that function. And for a piecewise-defined function, the input values of 𝑥 are defined by the subdomains of each subfunction.
For example, we’re told whenever our input value of 𝑥 is in the closed interval from two to six, our function is equal to two. Therefore, the input values of 𝑥 for our function are entirely defined by all of the subdomains. Since we can input any value of 𝑥 in any of the subdomains, the domain of our function will be the union of the subdomains. And this is true for any piecewise-defined function. The domain of a piecewise-defined function is the union of its subdomains.
So to answer our question, we need to find the union of these three sets. And there’s a few different ways of doing this. One way of doing this is to write inequalities for each of our subdomains. To say that 𝑥 is in the left-closed, right-open interval from zero to two is the same as saying that 𝑥 is greater than or equal to zero and less than two. Similarly, saying that 𝑥 is in the closed interval from two to six is the same as saying that 𝑥 is greater than or equal to two and less than or equal to six. And finally, saying that 𝑥 is in the left-open, right-closed interval from six to eight is the same as saying that 𝑥 is greater than six and 𝑥 is less than or equal to eight.
And our input values of 𝑥 can satisfy any of these three inequalities. But we can notice something interesting. In our first inequality, we want all of the values of 𝑥 less than two which are greater than or equal to zero. But in our second inequality, we want all of the values of 𝑥 greater than or equal to two which are less than or equal to six. So this is just all of the values of 𝑥 between zero and six. The union of these two sets is the closed interval from zero to six.
And in fact, we can see something very similar is true for the third subdomain. Our third subdomain includes all of the values of 𝑥 greater than six but less than or equal to eight. So this is then just all of the values between zero and eight inclusive, the closed interval from zero to eight.
Therefore, we’ve shown the domain of the piecewise function 𝑓 of 𝑥 given to us in the question is the closed interval from zero to eight.
