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CCSS.Math: ,

by now where you're used to seeing functions defined like H of Y is equal to Y squared or f of X is equal to the square root of x but we're now going to explore is functions that are defined piece-by-piece over different intervals and functions like this or you'll sometimes view them as ap SWAT or these types of function definitions they might be called a piecewise function definition and so let's take a look at this graph right over here this graph you can see that the function is constant over this interval for X and then it jumps up to in this interval for X and then it jumps back down for this interval or X so let's think about how we would write this using our function notation so if we say that this right over here is the x-axis and if this is the y is equal to f of x axis then let's see our function f of X is going to be equal to let's see there's three different intervals so let me give myself space for the three different intervals now this first interval is from not including negative nine I have this open circle here not a closed in circle so not including negative nine but X being greater than negative nine all and all the way up to and including negative five so I could write that as negative nine is less than X less than or equal to negative five that's this interval and what is the value of the function over this interval well we see the value of the function is negative nine it's a constant negative nine over that interval it's a little confusing because the value of the function is actually the also the value of the the lower bound on this interval right over here and it's very important to look at that this is negative nine is less than X not less than or equal if it was less than or equal then the function would have been defined at x equals negative nine but it's not we have an open circle right over there but now let's look at the next interval the next interval is from X is grid or negative five is less than X which is less than or equal to negative one and over that interval the function is equal to the function is a constant six it jumps up here sometimes people call this a step function it steps up it looks like stay years to some degree now it's very important here that at x equals negative five for it to be defined only one place here it's defined by this part it's only defined over here and so that's why it's important that this isn't a negative five is less than or equal to because that if you put negative five into the function this thing would be filled in and then the function would be defined to both places and that's not cool for a function it wouldn't be a function anymore so it's very important that this that it that when you input negative five in here you know which of these which of these intervals you are in you can't be in two of these in two of these intervals if you are in two of these intervals the intervals should give you the same value so that the function maps from one input to the same output now let's keep going we have this last interval where we're going from negative one we're going from negative one to nine from negative one to positive and I and X it starts off with negative 1 less than X because you have an open circle right over here and that's good because x equals negative one is defined up here all the way to X is less than or equal to nine and over that interval what is the value of our function well you see the value of our function is a constant negative seven a constant negative seven and we're done we have just constructed a piece-by-piece definition of this function and actually when you see this type of function notation it becomes a lot clearer why function notation is is useful even and hopefully well anyway hopefully you enjoyed that I always find these piecewise functions a lot of fun