 In this video, we present the solution to question number seven for practice exam number one for math 1210 In which case we're given the graph of a piecewise function and we're asked to find The formula for said piecewise function And so this question is intended to be a question for which we're gonna eliminate all but the correct answer So looking at this graph right here. It's important to identify. What are the pieces? It would look like I have three distinct pieces There's like three different lines that seem to be glued together And so as such I'll be looking for a formula that has three different pieces when you look at option a You have negative x and x minus four those are two lines I need three lines, so that's not gonna work when you look at option C You have a square root function and a line. That's two pieces and also I don't need a square root That's not gonna work Option e you have the constant function one the constant function negative one that those are linear functions But there's only two pieces. That's not gonna work When you look at option B Alright, you'll see that there is a line. There is a line. There is a line There's three pieces. So that seems that seems plausible Option D also has three pieces. So those are my contenders right now when you look at option D though You have a x plus one which is a line you have x minus five which is the line But the third piece x squared minus three x plus four that's gonna be the graph of a parabola We don't have any curvature on our graph right here, so it can't be option D So we're down to option B, but there's also the very dangerous answer f none of the above bump bump bump So even though we've eliminated everything else because there will be an option of none of the above Which is a legitimate answer right here We can't just accept B because we've eliminated everything else. So now that we have B in our focus. Let's check Okay, so we have three linear pieces. That's good. Look at their domains So this one will go from negative three to negative one For which negative three is right here negative ones right here Oh, okay The first line actually does go from negative three to negative one then the next domain will be negative one to one Which one would be right here? Okay, the second line does go from negative one to one That's so far so good. And then the last domain One to two we're gonna go from one to two. So okay, the domain matches up on this one That that's really promising notice if you went back to example D right here this one went from one to three Sorry, it's from negative infinity to one one to three and then three to infinity, right? So the domain wouldn't have agreed on this one even if we didn't rule it out for the parabola right there The domain is a very important part of this. Alright, so we have the correct domain. We have three linear functions Now let's verify those linear functions. Let's start with the middle piece Because after all I'm trying to see if I can eliminate it all if you take the line y equals x That'll be the diagonal line that goes through points like zero zero one one two two negative one negative one That's exactly this line right here So the second piece actually is totally kosher for this graph The next one to x minus one if we think of as a line in slope intercept form It's y intercept should be negative one at slope is two for which you look at this function right here You can see that the slope is to rise to over one What would the y intercept of this thing be if we continue in this regard you go over one down two The wire step would be negative one. Yeah, that's that's the right line there So this is very very promising The last one if we just want to be extra cautious right here three halves x plus one half Okay, so that tells me my slope is three over two So you go up three over two that is the correct slope if we were to continue with this slope we go up three Over two connect the dot right there Yes, sure enough that would be a one intercept of one half I would I'm totally on board that these are the three correct lines with the three correct domains. This is the correct formula for this piecewise function