# Question Video: Finding the Domain and Range of a Piecewise Function Mathematics

State the domain and range of the following function: 𝑓(𝑥) = −2 for −12 < 𝑥 ≤ −1, 𝑓(𝑥) = 11/2 for −1 < 𝑥 < 1 and 𝑓(𝑥) = 0 for 1 ≤ 𝑥 ≤ 4.

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### Video Transcript

State the domain and range of the following function. 𝑓 of 𝑥 equals negative two for negative 12 is less than 𝑥 which is less than or equal to negative one. 11 over two for negative one is less than 𝑥 which is less than one. And zero for one is less than or equal to 𝑥 which is less than or equal to four.

If we think of a function machine, we remember that the domain will contain all of the values that can go into the function and the range are all the values that can come out of the function. We have one function. And we’re given three conditions. The first line tells us what to do when 𝑥 is greater than negative 12 but less than or equal to negative one. This means that 𝑥 between negative 12 and equal to or less than negative one must be part of the domain. When 𝑥 falls in that domain, the output is negative two. Which means negative two must be part of our range.

Now, let’s consider the second set of information. We have the case when 𝑥 falls between negative one and one. This means we have an additional domain of 𝑥 between negative one and one. That also means that the range includes eleven-halves. This, of course, means that the domain includes one is greater than or equal to 𝑥, which is greater than or equal to four, and the range includes zero. We need to think about how we write this. We could write the range in set notation like this. The range is negative two, eleven-halves, and zero.

If we look closely at our domain, we’ll see that we have 𝑥-values beginning when 𝑥 is greater than negative 12 and ending when 𝑥 is less than or equal to four. This is true because the domain can include anything that can go into our function. And every value from something greater than negative 12 all the way until positive four has an output value for this function. And so, we say, for the given function 𝑓 of 𝑥, its domain is such that negative 12 is less than 𝑥 which is less than or equal to four. And the range includes negative two, eleven-halves, and zero.