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Current time:0:00Total duration:7:25

Worked example: domain & range of piecewise linear functions

CCSS.Math:

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 although the line changes depending what interval of X we are actually in and so let's think about its domain and then we'll think about its range so the domain of this just as a review the domain is the set of all inputs for which this function is defined and our input variable here is X just a set of all X values for which this function is defined and we see here anything anything negative 6 our lower our function isn't defined if X is negative 6 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's 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 six so this part right over here the low end of our domain is defined right over there so we said we could say negative 6 negative 6 is less than X and I'm leaving some extra let me just write here all real numbers actually let me write it this way X I could write it more mathy 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 let's think about the upper bound so as X goes now I just want to make sure that we fill in all the gaps between X being greater than negative 6 and X is less than or equal to 6 so let's see as we go up to an 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 to 4 were in this Clause up to an including 6 so X at the high end this has to be less than or equal to 6 less than or equal to 6 now another way to say this when kind of less mathy notation is X X can be any real number any real number such that such that negative six is less than X is less than or equal to six 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 this is a set of all inputs or sorry this is a 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 is X varies or as X can be any of the values in this interval what are what are the different values that G of X could take on so 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 going to worry about that in a second so when does this thing hit its low point so this thing hit its hits its low point when X is as small as possible and X is going to be as small as possible when it's approaching negative 6 so if X were equal to negative 6 it can't equal negative 6 here but if X is equal to negative 6 then this thing over here would be equal to negative 6 plus 7 would 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 of thinking about it if negative 6 is less than X then 1 is going to be less than G of X and the only reason I said that if you just put negative 6 into this negative 6 plus 7 is equal to 1 now this is going to hit a high end when it's as large as possible and 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's 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 for 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 want to make sure that makes sense to you because it's a little bit confusing because this takes on its minimum value when X is approaching or it's approaching it's a minimum value when X is approaching it's when X is approaching it's a maximum value because we're subtracting it so if you take the upper end even though this doesn't actually include 4 but as we approach 4 we could say okay 1 minus 4 is negative 3 so that's so G of X is always going to be greater than that and it's always going to be 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 2 X minus 11 it's going to hit its maximum value when X is as large as possible so its maximum value is 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 is going to be 1 it's actually going to be able to hit it because X can be equal to 6 and its minimum value is going to be when X is equal to 4 and it 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 it being greater 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 not shouldn't say it's defined all the way to I 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 of 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 include and including negative 3 and positive 4