OK, so there's one more group of functions that we want to talk about graphing, and those are piecewise functions. So here when we graph piecewise functions-- and by that I just mean a function that is defined in pieces. Pieces of the domain. So for example, here's a function f of t is equal to 3 if t is less than negative 5, t squared when negative 5 is less than or equal to t is less than or equal to 5, and t plus 1 for t greater than 5. So this is what I mean by a function that's defined in pieces. I have different definitions on different parts of the domain. Now, let's just do a couple of things. I just want to make sure. Example, find f of 3. So I just want to find the function value. If this is a function, it has one output for each input. So I should be able to get one output for this function. So finding f of 3, I go back up to this function and I find which of these three conditions, these second conditions out here are conditions on the domain. And I have to find where 3 fits into that piece. So is 3, when I plug it in for t, is 3 less than negative 5? No. So that is not the piece of the function that applies to my particular input. Now, is 3-- is negative 5 less than or equal to 3 less than or equal to 5? The answer is yes. 3 is between those two numbers. So this is the piece that I use for this input value, for the input value x or t is equal to 3. So when I find f of 3, I plug 3 into that particular equation, or that particular piece of the function, and I get an output value of 9. So that means on the graph I'll have the point 3 comma 9 in my graph. So I just want to make sure you really have to identify which part of the function your particular input should-- is applicable for your particular input and then use that one part of the function to plug in and to find the output value. There are not three output values for this, for t is equal to 3. So now we start talking about graphing this thing, it also occurs the graph happens in pieces. So I want to rewrite this problem down here. I'll rewrite the function since we kind of used this space. And maybe I'll try to write a little bit more compactly. t less than negative 5, t squared, and t plus 1 for t greater than 5. OK, So when I go to graph this, I really will have different pieces of the domain. And those split into how this graph looks. So let's 1, 2, 3, 4, 5. I want to get a little bit outside 5. And see what we have. OK, so now the first piece. Here's my splits at t negative 5. So I have a split here. And then again at 5. So on these different parts, if you can see I'm splitting the graph, this coordinate axes, into three different pieces. To the left of this first line, I'm going to have this function, f of t. Over here it looks like f of t is equal to 3 right here. That's a constant function. That's the same as y is equal to 3, which should be a horizontal line. So right here let's maybe put in this point negative 6, 3 like this. This should look like this. I should get an open circle there, because there's not really a point there. But otherwise it should look kind of like that. That's a horizontal line right there. Now, in this middle part of the graph, my function looks like t squared. And it behaves like t squared between negative 5 and 5, including the end points. t squared is a parabola. If you either plot using the vertex method or by points, you should get the following. 0, 0; 1, 2; 2, 4 and negative 2, 4. And then 3 is up here at 9 like this. So I'll graph this. I want to say this keeps going. I don't want you guys to be confused because I'm limited by space. This keeps going all the way up. At the point 5 I would be way up here. And this is the point 5, 25, right? So this keeps going up there and the same thing over here, negative 5 and 25. That is the point and this continues up until that point. So let's just imagine that that graph looks a little bit smoother than it does. I apologize. But that's kind of what this thing looks like. This parabolic shape between there. Now outside, when t is bigger than 5, my function looks like t plus 1 over in this region. So t plus 1 is just a graph of a line. You can either do this by plotting points. Sometimes that's easier. So when I plug in 6, I can figure out where I am. 6 in the output should be 7. And then it should kind of go-- I think I have a positive. At 5 I should get 6. So I should get an open circle there. 7 goes up. So you kind of get this line with slope one. That's going like that. And that continues forever. This is what the graph of this particular function looks like. This is a piecewise function. The graph happens in these, in this case, disjointed pieces. Now from this graph, I can also tell the domain is minus infinity to infinity. I'm allowed to plug in any value. And you can kind of see that from these conditions out here too. On the actual function I have t less than 5, negative 5. And that is a perfectly acceptable function to plug things into. A constant function doesn't have any restrictions on the domain. From negative 5 to 5, I also have a function that doesn't have restrictions, and t greater than 5. Now, you can also see here the range of this function starts here at this bottom part of the parabola. And this constant function will stay the same. Both the parabola and this line will continue to go up. Actually, the parabola ends at 25, but this line to the right will continue to go up to infinity. So the range here should be starting at 0 and going out to infinity as well. So this is a piecewise function that we get there. So let's try one more. I just want to do one more example to make sure. Piecewise functions are pretty important in the study of calculus and especially in limits that we'll be talking about in just a few days. So graph f of x is equal to x minus 1 when x is less than or equal to 0 and x plus 1 when x is greater than 0. So I want to go ahead and graph this function. OK. And again, here is the place. Let's mark the place where this changes, which is right here at x equal to 0, which is here on the y-axis. So I don't know if you guys can see this green being written over the black, but it's right there. I'll extend it a little bit top and bottom so you can kind of see. Right there on the y-axis, that's where my split occurs. To the left of the y-axis, I have a line with y-intercept negative 1. So I'm going to mark that here. Now, it's x less than or equal to 0 because this equals sign I fill in that y-intercept. That's the actual function value at 0. At 0 the function value should be negative 1. f of 0. When I plug that in, I look for 0 matches this criteria, and that should be equal to negative 1. And then this has a slope of 1, so I can go down 1 and to the left 1 to get more points and graph this. So it looks something like that. Now, the second piece of my graph has y-intercept of 1, but that's not the function value. If I fill that in, it's not a function because it fails the vertical line test there. So I have an open circle there and then otherwise it's kind of an imagined point. I go up 1, over 1 because this is also has slope one. And I get a few more points. And I graph this line to the right. This is what this function looks like. Again, domain minus infinity to infinity. Range, this line to the left keeps going down, down, down until we go to, I think, minus infinity beyond here. But this piece only goes up right here to negative 1. Then I have a gap and my piece on the right starts almost at 1, not quite, and goes all the way up to infinity. So range occurs in these two pieces, actually. Minus infinity to 1 including that 1. And then I start not quite at 1, but just a little bit up. And goes out to infinity. But I have this huge gap between minus 1 and 1 that's not in my range. There are no output values there. So it's not in the range of this function. So that's piecewise functions. Let me know if you have any questions. We will be using piecewise functions. I will ask you guys to graph them. And it kind of brings together a lot of the ideas that we do in graphing, so please practice. Otherwise, let me know if you have any questions.