Worked example: domain & range of step function

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 as a bit of review the domain is the set of all inputs for which our function is defined and over here our input variable is X so we could think about it's the set of all the values that X can take on and actually have this function be defined actually be able to 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 as X crosses 2 and is greater than 2 we fall into this clause and as we approach 6 but right when we get to 6 we fall into this clause right over here all the way up to an 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 at 0 or less the function is no longer defined as well so in order for this to be defined X has to be greater than 0 or we have to 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 than or equal to 11 it's defined for everything in between as we as we see once again as we as we get to to where here as we cross to between 2 & 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 may be all real values may be all right that way all real values such that such that 0 is less than X is less than or equal to 11 so now let's think about the range let's think about the range of this piecewise defined function and that's the set of all values that this function can actually take on and this one is is maybe deceptively simple because there's only three values that this function can take on it 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 to say this is a member of the set one five negative seven 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 to 1/5 or negative 7 this is maybe a little less pranaya little less a less Mathi way a less precise way of saying the same thing but one way or another f of X can only take on one of these 3 values