 In this video we provide the solution to question number six from the practice exam number one for math 1050. We are given the graph of a piecewise function f as illustrated right here. And we're supposed to find the algebraic formula that determines this piecewise function. Now be cautious on this one. None of the above is a possibility. And probably our best approach is to do this by the process of elimination. Now when we look at this function I can see there are three line segments that seem to build this function together. Therefore I can fairly safely eliminate any piecewise function that doesn't have three components to it. Alright so you'll notice like choice a this is a linear function this is a linear function but there's only one switch. It goes from negative one to two and then two to eight. I need three different pieces thus I need two switches for it so it can't be choice a there's not enough pieces. Same argument with choice C right here there's only two pieces. And in fact one of the pieces is a square root function these are straight lines it needs to look like something like mx plus b. So that's not going to work for choice C as well and I also can rule out choice E as a possibility. Notice this is one and negative one there's only two pieces to it. We need at least three pieces. Choice B is somewhat satisfactory because there are three pieces there. Same thing with choice D though so how do we eliminate between them? Well there's two approaches one could take at this at this perspective. You'll notice that each of these expressions for choice B look like a linear function some mx plus b. When you look at choice D you'll notice that there's a linear function linear function great but there's also this quadratic expression right here. This would suggest the graph would have some type of parabola inside of it. Now maybe you're not quite familiar with the shape of a graph based upon these formulas lines and square roots and parabolas. Maybe we're not familiar with that enough and that's okay look at the domains look at the switching numbers. The one line switches at negative one and it switches at the other positive one right. And so we have to look for a function which is using those switching numbers. You'll notice that for B it switches at negative one it switches at one right. There's one and negative one there it switches at those values that's pretty good. When you look at choice D it does switch at one but the other switching numbers at three. So just considering the domain of these functions even if you don't understand how to interpret the formulas yet that's okay. We can rule out choice D as a possibility. Now like I said this none of the above is somewhat of a dangerous thing on this question. It's understandable. If you continue to look at the domain of choice B right we end at negative three over here that's included and then two over here. X equals two. So the domain is spot on on this thing and so the only concern might be the functions for which we can use some test points to help us out here. Like if I plug in let's say negative one into this expression just get negative one. That's this point. If I plug one into this expression I get one right here like so. If I plug one into this thing you're going to get two minus one which is one. Again that point they're touched no big deal. If you plug in two you'll get four minus one which is three that gives you this point right here. And then lastly if you plug in negative three into this one you'll get negative nine halves plus one half which is negative eight halves which is negative four. Which is this point right here. So you can see that all of these points seem to match up here. And so even if you have a little bit you're a little bit shaky on how to graph linear or quadratic functions. We can actually with very strong confidence suppose that B is the correct answer. Now of course if you are if you do not a graph a linear function you can see that that's exactly what these things are you can look at the slope. What have you but it's not actually necessary to rule this one out. B is in fact the correct response for question number six.