Current time:0:00Total duration:7:25
0 energy points
Studying for a test? Prepare with these 2 lessons on Absolute value & piecewise functions.
See 2 lessons

Worked example: domain & range of piecewise linear functions

Video transcript
So we have a piecewise linear function right over here for different intervals of x. g of x is defined by a a line or the line changes depending what interval of x we're actually in. And so let's think about its domain, and then we'll think about its range. So the domain of this, this is a review. The domain is the set of all inputs for which this function is defined, and our input variable here is x. This is a set of all x values for which this function is defined. And we see here. Anything, anything negative 6 or lower, our function isn't defined. If it, if x is negative 6 or or lower than that. I don't -- it doesn't, it doesn't fall into one of these three intervals. So there is no definition for it. It doesn't say hey do this in all other cases for x. It is just saying, look, if x falls into one of these three conditions, apply this. And if x doesn't fall into one of those three conditions, well this function g is just not defined. So, to fall into one of these three, you have to be at least greater than negative 6. So this part right over here, the low end of our domain is defined right over there, so we say, we could say, negative 6 is less than x and I'm leaving -- so let's write it here. All real numbers -- actually let me write this way x, I could write it more math-y. I could say x is a member of the real numbers such that, such that negative 6 is less than x. Negative 6 is less than x and I also think about the upper bound. So as x goes, I just wanna make sure that we fill in all the gaps between x being a greater than negative 6 and x is less than or equal to 6. So let's see. As we go up to and including negative 3, we're in this clause. As soon as we cross negative 3, we fall into this clause up to 4, but as soon as we get 4, we're in this clause up to and including 6. So x at the high end is said to be less than or equal to 6, less then or equal to 6. Now another way to say this and kind of less math-y notation is x, x can be any real number, any the real number such that, such that negative 6 is less than x is less than or equal to 6. These two statements are equivalent. So now let's think about the range of this function. Let's think about the range, and the range is, this is the set of all inputs , oh sorry, this is the set of all outputs that this function can take on, or all the values that this function can take on. And to do that, let's just think about as x goes, but x varies or x can be any values in this interval. What are the different values that g of x could take on? Let's think about that. g of x is going to be between what and what? g of x is going to be between what and what? g of x is going to be between what and what? And it might actually, this might be some equal signs there but I'm gonna worry about that in a second. So when does this thing hit its low point? o this thing hits, hits its low point when x is as small as possible. An x is going to be as small as possible when x is approaching negative 6. So if x were equal to negative 6, it can't equal negative 6 herer but if x is equal to negative 6, then this thing over here would be equal to negative 6 plus 7, would be, would be 1. So if x is greater than negative 6, g of x is going to be greater than 1, or another way to think about it is if negative 6 is less than x, then 1 is going to be less than g of x. And the reason I said that is if I put negative 6 into this, negative 6 plus 7 is equal to 1. Now this gonna hit a high end when it as large as possible. The largest value in this interval that we can take on is x being equal to negative 3. So when x is equal to negative 3, negative 3 plus 7 is equal to 4, positive 4. And it can actually take on that value because this is less than or equal to, so we can actually take on x equals negative 3 in which case g of x actually will take on positive 4. So, let' do that for each of these. Now here we have 1 minus x, so this is going to take on its smallest value when x is as large as possible. So the largest value x can approach for, it can't quite take on for, but it's going to approach for. So if x, let's see, if we said x was 4, although that's not this clause here, 1 minus x, 1 minus 4 is negative 3. So as long as x is less than 4, then negative 3 is going to be less than g of x. I wanna make sure that makes sense to you because it can be little bit confusing because this takes on its minimum value when x is approaching, or it's approaching its minimum value when x is approaching its, when x is approaching its maximum because we're subtracting it. So if you take the upper end, even though this doesn't actually include 4, but as we approach for, we could say, OK, 1 negative 4 is negative 3 so that's, so g of x is always going to be greater than that, as well it's going to be going to be a less than. Well what happens as we approach x being equal to negative 3? So, 1 minus negative 3 is going to be positive 4. So this is going to be positive 4 right over here. And these are both less than, not less than or equal to because these are both less than right over here. And now let's think about this right over here. So 2x minus 11 is gonna hit its maximum value when x is as large as possible. So its maximum value's going to be hit when x is equal to 6 So 2 times 6 is 12, minus 11. Well that's going to be 1. So its maximum value's going to be 1. It's actually going to be able to hit because x can be equal to 6. Its minimum value is going to be when x is equal to 4, and actually can be equal to 4. We have this less than or equal sign right over there. So 2 times 4 is 8, minus 11 is negative 3. So, g of x in this case can get as low as negative 3 when x is equal to 4. So now let's think about all of, all of the values that g of x can take on. So we could say, we could write this a bunch of ways, we could write g of x is going to be a member of the real numbers such that -- let's see. What's the lowest value g of x can take on? g of x can get as low as negative 3. It can even be equal to negative 3. This one just has been greater the negative 3, but here can be greater than or equal to negative 3. So negative 3 is less than or equal to g of x, and it can get as high as, it can get as high as Let's see. It's defined all the way to 1 and then -- or I shouldn't say it is defined all the way to 1. It can take on values up to 1 but it can also take on values beyond 1. It can take on values all the way up to including 4 over here. So it can take on values up to and including 4. So g of x is a member real numbers such that negative 3 is less than or equal to g of x is less than or equal to 4. So the set of all values that g of x can take on between, including and including negative 3 and positive 4.