 Welcome to the review for Exam 1 for Math 1050 College Algebra at Southern Utah University. As usual, I'll be your instructor today, Dr. Andrew Misildine. This is our first exam for this course, and so there's a lot that we need to talk about, a lot of unknowns. I should first start off and say that I can't say everything about this exam in this video, because there are some specific things about the exam that do change from semester to semester, like the time, the place, the manner in which the test will be administered. Is the test in class? Is it remote? Is it in the test center? Those things do change depending on the section, depending on the semester and such. Those won't be found here. That semester-specific information like what days of the exam, what is available, what am I allowed to use on the exam, whether notes, calculators, that type of stuff. You want to ask your instructor, aka me, in person, or you can find the information on Canvas. So what won't change from semester to semester is the specific structure of the exam, the topics covered. This first exam is going to cover from our lecture series, Lectures 1-8, which comprises what the lecture notes call Unit 1, Chapter 1 of the Fundamentals of Functions. So from our lecture series, we're talking about Lectures 1-8. Like I said, this would be Section 1.1 through 1.6 if you follow the numbering that's in the lecture notes and in the accompanying videos, right? And so just as a reminder, we're talking about the basics of functions, like evaluation of functions, determining the domain and range of functions, the vertical line test, the horizontal line test, the combining functions, piecewise functions, composition, transformations of functions, symmetry, inverses of functions, all of that type of stuff we had talked about in our very first unit. This exam will consist of two types of questions. The first question is the multiple choice questions, for which in the multiple choice, which we'll see some examples of those in just a second, the multiple choice questions will provide you a question. And then there could be like four, five, six, maybe more options for which you then are expected to select the one single correct option. Clearly indicate that there won't be, you won't select more than one, you just select the one that is correct based upon the list that you'll see. And we'll see some examples of those in just a second. The multiple choice questions for this exam, there is officially 11, although there's one that's in the other section, it's sort of like multiple choice, I'll explain what that means in just a second. But you have 11 multiple choice questions on this exam. They're worth each five points out of a hundred. And when it comes to the multiple choice questions, there's no partial credit. You either select the correct answer to get the full five points, or you don't select the correct answer, or you select too many answers, or you don't select anything, something like that. If you don't select the single correct answer, you don't get the points. So each of those is an all or nothing situation. The second section on the test is what we call the free response section. This will consist of questions 12 through 16. These are mostly worth 10 points each, but they're not all the same pointage. For example, question 15 is a little bit on the easier side, it's eight points. And then question 12 is actually only worth five points. It's actually on par with the multiple choice. Like I said, it kind of is a multiple choice question, but the nature of the question doesn't lead itself to selecting the best option. You actually provide yourself and I'll explain what that means in a second. This very last question is of course for the notes that you're allowed for the exam, two points for that, a pass-fail type of thing. Again, send me an email or ask me in person if you have any specific questions about what is required on this one, because that might change from semester to semester. So don't worry about that so much. All right, so with that in mind, I want to spend the rest of this video talking about the specific types of questions we're going to see on this first exam, understanding that the examples we see in this practice exam will not be exhaustive. There could be other types of questions one could see, and I actually want to point out some comments about those as we go forward. So let's take a look. Question one in the multiple choice section, remember, these are each five points each. This first question is really three questions for the price of one, right? You'll be given the graphs of three equations, right? We'll just say you're given three graphs. And I say graphs because they're not necessarily functions. That's actually the nature of this question. Which of the following graphs coincide with functions? And so we see there's option I, option one, option two, option three, right? And so you have to determine which of these three is a function. You maybe think that one and two is a function for which you would then select choice D, which is one and two. Maybe you think two and three are functions, in which case you would then select F. Maybe you think they're all functions for which you choose G. Maybe you only think two is a function so you select B. Or lastly, maybe you don't think any of them are functions. You can then select choice H would be none of them. So any of the eight possibilities is an answer here. Select them accordingly. On this question right here, as we're given the graphs of these relationships and we want to determine if it's a function relationship, the tool you probably want to use is the vertical line test, which we learned about of course in lecture one specifically section 1.1. All right, so apply the vertical line test to determine which of these things are in fact functions or not. Clearly the three graphs you see on your version of the test are open to change, but the otherwise the format of this question will be similar to what you see just right here. Moving on to question number two, you'll be given a function expressed algebraically on the practice test. We have this quadratic function x squared plus three x plus four. And you're asked to evaluate the function. All right, not much more to it than just that. You'll be given an algebraic function, evaluate clearly the exact function you have to evaluate with my change. The number you're going to evaluate on of course is prone to change, but I want to see if you can algebraically evaluate a function. And I say algebraically, but you could potentially have a numerical representation or a geometric one, but most likely this will be an algebraic evaluation of the function, much like we did in lecture three in particular section 1.2 according to the lecture series there. If you want some more practice, turn to that lecture number three there. Question number three, you'll be given the graph of a function. So this again, while it could be a numerical representation, that's a possibility. An algebraic one it definitely won't be. Most likely though, it will be a geometric representation of the function. You'll be given a graph of the function in some regard. And you'll be asked to compute maybe the domain of the function or also the range of the function. This question is asking for domain, but a possible variant could be, here's a graph, what's the range? The two questions are related, but not exactly the same thing. We did these things day one in lecture one from section 1.1 of our lecture series. We talked extensively about representing functions numerically and graphically and we identified how to find their domains and ranges. You might be noticing in this video, I'm not providing the solutions to these questions. I'm doing that for a reason, of course. This review is just to tell you the types of questions you might see and the practice test provides a sample of what you might see. I don't want to provide the answers in this video so that you have the opportunity to try these on your own without me giving all the answers away. Now, the document that you have this practice exam, hopefully you have in front of you, if you don't, I'd say pause the video, download it from Canvas and follow along. That same PDF that has the practice exam also has solutions at the very end plus also there's a video solution for each and every one of these practice questions that can also be found on Canvas. So I'd turn your attention to that if you do want to see solutions to these questions. This video just focuses on the question types that you will see. Question four is going to be about the arithmetic combination of functions. So like do we add functions, subtract functions, multiply functions, divide functions? How does that affect the formulas of the function? How does that affect the evaluation of the function? So this one right here, we're given three functions f, g, and h. You'll probably always be given three functions. And then you'll be asked in this specific one, I want you to find what's the difference of f and g evaluated at negative one. So there's an evaluation there, but also how do you combine functions together via subtraction in this example? But you could also do addition, multiplication, division. Those are those arithmetic combinations of functions. How do you do that algebraically? How do you do that numerically? How do you do that geometrically? Now most likely this question will be formatted as an algebraic function type question. But it could be, it could be a variant could be to give you the graphs of functions, or that one's a little bit on the harder side, but a numerical representation is actually fair game for something like this. If I give you three functions who are given as tables, could we find the difference? Oh sure, we probably could. And so this arithmetic combination was first mentioned in lecture four for the algebraic version, I should say, in particular section 1.3. The numerical and geometric ones showed up a little bit earlier. I believe in lecture two is where that emerged section 1.1. But 1.3 is probably where you want to go to see some more examples. And of course, also go to the corresponding homework problems as well to get some extra practice on these things. Moving on to the second page of the multiple choice test for now on question five. It looks a lot like question one, that is you're given three graphs and you're supposed to determine something about those three graphs right here. This time you're asked to determine which of these graphs is a one to one function. So the first thing to remember here is that if it's a one to one function, it still needs to be a function, right? If my wife tells me to go out, go to the pet store, bring home a white puppy, and then I come home with a white snake, right? That Python is white, that satisfies the adjective, but if it doesn't satisfy the noun, it's kind of a problem still, right? So it needs to be a function. So the graph first needs to pass the vertical line test to be a function. The additional adjective of one to one means that the function needs to also, the graph, I should say, passes the horizontal line test. So in question one, we just need the vertical line test to be a function. On question five, we need both the vertical and horizontal line test to be a one to one function, and then how you choose your answers similar to question number one. Question number six is going to be a question about piecewise functions. I should say before going on to six though, on question number five, we talked about one to one functions in lecture seven, which was about inverse functions. That was section 1.6, according to the lecture series. All right, question number six. This is going to be a question about piecewise functions. You'll be given the graph of a piecewise function, and then you're supposed to determine what is the algebraic formula that gives you that piecewise function. Five options will be provided, and then the sixth option is the most deadly of all, none of the above. I'm not typically a big fan of none of the above because when students get stuck, they often turn to none of the above. That's sort of the default answer, which I should mention that none of the above is not any more likely the answer than any others. So if you're trying to choose one randomly, honestly, any of the choices are potentially better. But I would hope that even if you struggle on this one, you could eliminate some of the answers, because that's really how this question is meant to be solved by a process of elimination. That is, can you eliminate some obvious ones over other ones, right? So some things to note here, it's like how many pieces does the piecewise function have? When you look at the domain, this one has two pieces, this has two pieces, this has two pieces, this has three pieces, this has three pieces, but then there's, of course, the scary one. And then also, what do the things look like? Well, we have a linear function, linear function, it's a line, that's a square root, right? That has some curvature to it. Line, line, line, line parabola, again, there's some curvature to it. Line, this is just constants. So the idea is by the process of elimination, I do believe you could then determine which of these, you know, this graph coincides to which formula, what have you. I'm not saying it's the first one there, just you would then pick one based upon that. And if you really feel like none of them match up, then select, then select F. If you really think it's like, I think it's either A or F, you're probably better off going with A there. That would be my guess as the test writer, but that's not to say none of the above never shows up. But if the answer was none of the above, you probably could determine from the graph that none of the other five could possibly work. Piecewise functions we introduced in lecture three when we started talking about algebraic functions there. And so turn to section, excuse me, 1.2 from the lecture series, lecture three, for some more practice there. Question number seven at the end of the second page is going to be about function composition, for which you're given two functions. In this case, F and G, they're given algebraically. What is the composite of the two? So if you compose the two functions together, so remember this is the F of G of X, you put one function inside the other. What is the composite formula? Remember order of operations here matter, that F of G is not the same thing as G of F of X. The order does in fact matter, so make sure you do that. Can we come up with a formula? Can we do just an evaluation? Can we do F of G at two or something like that? And this is, of course, given algebraically, like all these other examples, this question could be swapped into a numerical representation of the function. It could be switched to a geometric one, or maybe it could be a combination. One of the functions could be given numerically, but the other one's algebraically, and then you're asked to compute F of G of seven. Could you do that? Ideally, you should be able to. Now, in practice, though, the questions that is the software I use, written by myself to generate these questions, they very much mimic the versions you see in the practice test, but I should mention that there are some variants that might occur that if you're not paying attention to this video, you might have otherwise not thought about, such as the one I just mentioned a moment ago. Function composition was given in lecture four, and our lecture series particularly will be labeled as 1.3 in the lecture notes. Moving to the last page of the multiple choice section, we're on to question eight now. Question eight, you'll be asked to compute the average rate of change of a function on an interval. Expect the function to change, expect the interval to change, but other than that, you'll be asked to compute the average rate of change here. And again, this could be given numerically. I do like to do geometric representations too, like you can see question 11 down here, but the graphs typically take a lot of space on the page. So from a formatting point of view, the algebraic and the numerical ones typically take less space. So you can anticipate many of these calculations will be more in line with that, like you see right here. Of course, the numerical calculation would be a lot easier than the algebraic one, because the algebraic one, we actually have to figure out what is f of such and such. The table, we can just read it off. So in some respect, we might like the table approach, but be prepared for all of these versions. Rates of change, average rates of change were given in lecture six, and according to the lecture numbers, this was what was labeled section 1.5. Question number nine is another question about function composition, but this time we're trying to do function decomposition. That is, you're given the combination of the two functions. So capital F of x, which is not the same thing as little f of x, mathematics is case sensitive, you're giving capital F of x, and you're told that capital F is the composition, the composite of f and g, for which this case little f is given the outside function, and you're asked to determine what is little g? What is the inside function? So what little, what inner function can be combined with the outer function, which is given to produce the whole function? So how do you decompose it in that regard? So that, of course, happened in like, we talked about that in lecture four, again, section 1.3 by the lecture numbering was about function composition. At the very end of that lecture, we talked about function decomposition. It's a very important principle for calculus. Honestly, a lot of these questions are average range of change, very important for calculus. Decomposition is super important for calculus. So you'll be asked to do this on this test. It's a very important question. Then we get to lecture 10, excuse me, question 10, for which much like questions 1 and 5, you'll actually be given three functions. This time, they will be given algebraically, and you'll be asked to determine symmetry of these things. This question asks, which of the following are odd? You could also be asked, which of the following are even something like that. That's a variation. Of course, the three functions in play, f, g, and h will probably change. And the basis is the same, right? You want to determine, well, maybe f is odd. I should do a check mark there. Maybe g's not, maybe h is not, so you select a. But then you're like, well, maybe h is, in which case, then you go and select e. Same thing like questions 1 and 5 right here. To test for symmetry, you're going to be wanting to look at f of negative x. Because if this turns out to just be f of x, then we, in fact, we have an even function. This version of the question is not asking for that, but it could come up. Maybe you could be asked to find even functions. But also, if f of negative x equals negative f of x, that actually is an odd function, which is what you're looking for on this question. Or if neither of these things happens, then neither of the two symmetries are present. So you want to know how to test for symmetry algebraically. That showed up in lecture 5, at the very end, as we talked about, graph transformations. We introduced geometric symmetry in lecture 2, but then the algebraic version showed up a few lectures later in lecture 5. So now we get to the last question in the multiple choice section. This one will be given graphically, in which case you'll be asked something, given the graph of a function, where is it increasing? That's a possibility. Where is it decreasing? So some questions about monotonicity there. Are there any local extrema? Like identify, are there any local maxima? There could be more than one. Are there any local minima? There could be more than one. What's the global or absolute maximum? What's the global or absolute minimum? So questions about monotonicity could come up on this. But we could also ask questions about concavity. On what intervals is the function concave up? On which intervals is the function concave downward? We could ask that question as well. And then we could ask about where are the points of inflection? These are the points where it switches concavity. These topics was the main topics from lecture 2 in section 1.1. So be prepared to do that. Most of these will be given, will be intervals, and thus should be written in interval notation. When you're asking on what intervals is it increasing, decreasing, or what intervals is it concave up, concave down, we're asking about the x-coordinates. So these right here, like this example, negative 6 to negative 2. So something like, ooh, negative 6 to negative 2. And then 1 to 6. These are x-coordinates. This is a domain for the x-coordinates. So that's what you're selecting, not the y-coordinates. That's a common mistake on these type of questions. So now we move on to the free response section. The first question, 12, is only worth 5 points, the same amount as a multiple-choice section question. In this one, you'll be given the graph of a function f. This function will be 1 to 1, and then you're asked to sketch the graph of f inverse. And so that's all that one has to do. For which the multiple-choice questions are inherently meant to be on the easier side, not as many involved steps as opposed to the free response. In the free response section, I don't think I mentioned this earlier, but in the free response, you must show all of your work to get full credit. You can get partial credit for partially correct work, but if you only put the final answer, even if that answer is correct, you will receive little, maybe no points whatsoever. You need to show all of your work. Now the question number 12 here, there's really not a lot of work to show because to find the inverse graph, you just want to reflect the given 1 to 1 graph. You want to reflect it across the line y equals x, which of course is the diagonal line y equals x, something like that. You want to reflect the graph across that line. And so there's really not a lot of work to show here. And so this would just be based upon what your final answer is. That's why I'd say it's really more like a multiple-choice, but if I listed six options, that would take up so much room on the page, you'll just be asked to produce one yourself. All right, so question number 13 is more in line with what we expect from a free-response-type question, for which you'll be given a function f of x, which equals, in this case, you have this linear fractional negative 3x minus 4 all over x minus 2. And you're asked to compute f inverse of x. So this is the situation, remember, where you have like y equals f of x, and then you're going to swap the rules of x and y. So x and y swap, so you get x equals f of y, and then you do a bunch of algebra until you eventually get to something like y equals f inverse of x. And you should be explicit that, oh, I've discovered f of x, or f inverse of x, here it is, something like that. So that's the calculation here. Inverse functions, we talked about in lectures seven and eight, both coincide with the numbers 1.6 from the lecture notes. Graphing inverse functions, like question 12, that was in lecture seven. Solving for the inverse is algebraically like question number 13 here. That is what I was found in lecture eight. Of course, the exact functions you find, you should expect those to change, but question number 13 will otherwise ask you to find the inverse of a function algebraically. We're now to the last page of the exam, question number 14. We'll be asked to evaluate and simplify the difference quotient f of x plus h minus f of x over h, where the function f of x will change. So the idea here for this difference quotient is you have to take f of x plus h. What does that mean? x plus h means you would substitute in x plus h for every occurrence of x inside of the function. So if I had something like, say, f of x is the square root of x plus 1, then that means f of x plus h would mean we replace each of the x's with an x plus h, like so. So it does not mean that we just add h to the function. That, it doesn't mean that. That's a no-no, because basically the last assumption is just assuming that f of x plus h is equal to f of x plus h, which I get it. When you say that loud, that's what it sounds like, but that is a falsehood. That's suggesting that the vertical zone and the horizontal zone aren't the same thing. And I wouldn't do the twilight zone noise if I suggested it was just like real life. No, the two behave very differently. Adding an h inside of the parameter is very different than adding it outside of the function. And that needs to be recognized here. Because let me point out to you that if x plus h just meant you added h to the formula, then you would always get something like the following, which this following work here is erroneous. I just want to point this out here. You get f plus f of x plus h minus f of x, for which the f of x's would cancel out. You end up with h over h, within simplifies to be 1. In which case, notice I never used anything about f of x. This would be true in general that, oh, the difference quotient's always simplified to be 1. Here's the truth of it. No, they don't. If they always simplified to be 1, why would we talk about them if it always simplified? No. The thing is, they don't generally simplify to 1. If you get your answer as 1 at the end on question number 14, I can tell you that 999,999 out of a million, you are wrong. It's not going to simplify to be 1. It should be simplified into something else. It's an evolved process. Do pay attention to these type of questions here. Difference quotients we talked about in detail in lecture 6, which coincides with section 1.5 from the lecture notes there. Be prepared. This is probably arguably the hardest question on the test, which is why it's worth the most points, 10 points there. Be prepared. While the function will change, you will have to simplify a difference quotient here, and it won't be 1. Question number 15, we're almost at the end here, second to last question. This one's a little bit on the easier side compared to some of the others, so it's only worth 8 points here. You're asked to find the domain of a function. It'll be given algebraically. The problems we want to look out for here is if you divide by 0, that's a problem. So if your function involves fractions of any kind, check to see if any of the denominators are 0. And if any of the denominators contain a variable like x, like we do in this one, then you might have to solve an equation. Go about do so. We cannot allow the denominator to go to 0. Also, we don't want to take any square roots of a negative. Odd radicals are fine. Like the cube root of a negative, totally fine. The fourth root of a negative, no, no. The fifth root of a negative, kosher. The sixth root of a negative, no, no. The idea is even radicals, square roots, fourth roots, sixth roots, when you take negatives of them, you're going to produce imaginary numbers for which at this point in the course, we are pretending they're not real, all right? For odd ones like cube roots, fifth roots, that's never a problem. So like you see on this one, oh, there is a division, so there could be division by 0. There is a square root, so we could be taking the square root of a negative. So then those are the problems we have to investigate. Dividing by 0 leads you to solving certain equations. The square root of a negative solves, you have to solve some certain inequalities. So look for those type of issues and then report your answer in interval notation. There's not typically a lot of work that one shows on question 15, but there is some work. So do show everything you can. All right, question number 16. This is the last question, the last real question on the exam, I should say. And you're asked to graph a function and we're going to graph it using transformations. So you're expected to graph the function f of x, which will be given here for this example. You're supposed to do negative 2 times the absolute value of x minus 2 plus 3. Now, if you don't know what the absolute value function looks like, because you're supposed to graph this using transformations from the basic graph, y equals the absolute value of x, no big deal. The basic graph is given to you. That's what you see on the screen right now. This is the standard non-transformed absolute value function for which this could change, this could be a parabola, this could be a lot of different things. Don't worry about it, it will be given to you. What you have to do is identify what transformations were given. So like, what does this negative 2 do to the function? What does this negative 2 do to the function? I'll give you a hint, they do different things. What does this plus 3 do? There are transformations you should be looking for. Like, are there reflections across the x or y axes? Are there vertical stretches or horizontal stretches or compresses that are occurring on the graph? Are there horizontal or vertical shifts, translations up, down, left, right on the graph? These you would list to the side. That's part of showing your work. You need to indicate all of those. If you have the correct graph over here, it's not this picture. Like, if you're like, oh, the graph looks something like this. If this is correct, but you do not list the transformations, we're going to get a big, fat zero. The graph is necessary for full credit, but in order to get credit from the graph, you have to list the transformations. The graph without the transformations, even if correct, is worth nothing. The points for the graph are contingent upon these transformations. All right, so you list the transformations correctly. You can get five out of the 10 points. Then your graph needs to agree with the transformations you said. So there's another five points available there. Also, another thing to point out here is you need to label three of the points on the graph. So you'd say something like, oh, assuming this is the final answer, it contains zero, zero, three, three, negative four, four, something like that. Just label three points. And this is to help us when we struggle, perhaps with the not the best drawing skills. Labeling the points makes it a lot more clear what it is you intended. So that is necessary for full credit as well. Graph transformations was, of course, the main topic from lecture five, which in the lecture series was labeled section 1.4. So go there for more practice on these type of questions if you need them. I didn't mention this with the domain one, but finding the domain algebraically, that is the domain convention, that came from lecture three, which was section 1.2. So I'll throw that out there. So that brings us to the end of the test. Just these 16 questions. If you do have any, if you have questions as you're studying bio means, please let me know. I'd be glad to help you. Many of the resources you have available on Canvas, of course, is this practice test with the video which you're watching now. I mentioned already that there are detailed solutions to each and every one of the questions in this practice test. The PDF document has solutions, and there are also solution videos you can find on Canvas. There's also an exam syllabus that's available on Canvas that goes through very similar to what this video just did and talks about what are the important things you need to know for this test as you're studying for it. All right. So that brings us to the end of our review. Thanks for watching. And I'll say one more time, if you have any questions, don't hesitate to ask me. I'm here to help you do the best you can. Just let me know what I can do. And I will see you next time, everyone. Bye.