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Current time:0:00Total duration:2:44

Worked example: domain & range of step function


Video transcript

I have a piecewise defined function here, and my goal is to figure out its domain and its range. So first let's think about the domain. And just a bit overview, the domain is the set of all inputs for which our function is defined. And over here, an input variable is x, so to think about, it's the set of all the values that x can take on, and actually have this function be defined, and actually figure out what f of x is. And when we look at this, we see, okay, if 0 is less than x is less than or equal to 2. We're in this clause, it's x crosses 2 and it is greater than 2. We follow this clause As we approach 6 but right when we get to 6, we fall into this clause right over here, all the way up to and including 11. But if we get larger than 11, the function is no longer defined. I don't know which of these to use. And if we're 0 or less, the function is longer defined as well. So in order for this to be defined, x has to be greater than 0 or if we say 0 is less than x, and you see that part right over there. And x has to be less than or equal to 11. and x has to be less then or equal to 11. It's defined for everything in between. As we, as we see, once again, as we get to 2, we're here. As we cross 2, between 2 and 6, we're here, and at 6, from 6 to 11 we're over here. So we're defined for all real numbers in this interval. So our domain is -- actually let me write this all, all real values, are all real all real values. maybe -- Let me write that way. All real value such that, such that, 0 is less than x, is less than or equal to 11. So now think about the range. Let's think about the range of this piecewise defined function. And that's a set of all values that this function can actually take on. and this one is, is maybe deceptively simple because there're only three values that this function can take on. You can take on, f of x can be equal to 1. It can be equal to 5, or it could be equal to negative 7. So the range here, we could say that f of x needs to be a member of, this is just a fancy mathy symbol, just to say this is a member of the set 1, 5, negative 7. f of x is going to take, is going to take on one of these three values. Another way to say it is that f of x is going to be equal 1, 5 or negative 7. This is maybe a little less -- a little -- a less a less mathy way, a less precise way of saying the same thing. But one way or another, f of x can only take on one over these three values.