Video: Determining the Domain of Piecewise-Defined Functions

Find the domain of the real function 𝑓(𝑥) = 𝑥² − 19 if 𝑥 ≤ 6 and 𝑓(𝑥) = −19 if 6 < 𝑥 < 23.

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Video Transcript

Find the domain of the real function 𝑓 of 𝑥 is equal to 𝑥 squared minus 19 if 𝑥 is less than or equal to six and 𝑓 of 𝑥 is equal to negative 19 if 𝑥 is greater than six but less than 23.

So the first thing we need think about is what is the domain of a function. Well the domain is the set of all possible independent values. So it’s all possible 𝑥 values. It’s where the function exists. So the first place that we’re told that function exists is when 𝑥 is less than or equal to six. Well as we’re told that 𝑥 is any value less than or equal to six, then it means that 𝑥 is any value less. So therefore we can say that our lower bound is going to be negative infinity. And when we show it on our notation, we have a parenthesis, not a bracket. And that’s cause a parenthesis means that it’s not including. And our value of negative infinity, it means that it’s not including negative infinity cause negative infinity isn’t an 𝑥 value itself. It just tells us that 𝑥 can take up any value in this case less than or equal to six.

The next part of our function tells us that 𝑥 is greater than six but less than 23. Now if we check out both parts of our function, we can see that the breaking point for the piecewise function includes six. So therefore, there aren’t gonna be any holes in this function. So that means there will be possible values of 𝑥 all the way up to, but not including, 23. So again, I use the same notation. So we’ve got a parenthesis because it’s not including the 23. So therefore, the domain of the real function is as shown here. We’ve got negative infinity, 23 inside our parentheses. And that’s because there is an outcome for every 𝑥 value between negative infinity and 23, but not including negative infinity and 23.

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