in this video I will do the review of the topics that I collectively refer to as pre-calculus and the main objective for the viewer is to understand the place of three calculus in the landscape of lower division math courses and to be able to identify the main subjects of interest and tasks learned in precalculus now previously I have uh shown you this layout or what I call the landscape of lower division math courses and you can see that the pre-calculus calculus one calculus two form this sort of pipeline from the beginning of mathematics into the large field of lower division math courses and pre-calculus sits right at the beginning of this Pipeline and it is naturally a a prerequisite for calculus one so coming out of pre-calculus you have to be ready for calculus one but also pre-calculus uh serves uh as a collector or as a merger of two branches um one being geometry trigonometry and the other one being arithmetic P algebra algebra so somehow in pre-calculus we have to develop a unified approach to uh previously learned math and as you may already know uh that the main subject of interest in pre-calculus is uh functions so in this video uh or in pre-calculus we learn functions and this what we call pre-calculus review will actually end up being a functions review and to start we have to first be able to list all different types of functions and you can pause the video here and try to list them all out and I will reveal all of those functions in a second so here is the list of all types of different functions that we learned coming out of pre-calculus um here you see your familiar eight uh functions coming out of algebra branch and here we have a trigonometric functions uh and there is many of them there are 12 of them but collectively we'll just refer to them as trigonometric functions and what does it mean to know each of these types of the functions so um here's what I'm calling a knowledge Matrix so every piece of knowledge that we acquire about functions will belong somewhere in this Matrix and um the rows Above This functions line uh labeled as operations expressions and equations and I did A Brief Review of arithmetic in which I discussed all operations that are out there and in that review I think I said that operations are predecessors of functions and now we kind of see how exactly they um become functions eventually and the next review I did was on pre-algebra in pre-algebra we learned uh how to handle each of the different types of expressions and guess how many Expressions types are there and there are exactly eight of them and the names are exactly the same as the names of functions so naturally you have to be able to work with Expressions before you learn any of the precalculus uh also I think I made the point that expressions are building blocks uh meaning that we use Expressions to create equations and you already know that we also use Expressions to Define functions but equations I discussed briefly in the video called algebra review so in that video we talked about different types of equations and what is there we do with them we solve them we also discussed inequalities and systems there but the those will come up uh eventually but the biggest objective was to learn how to solve equations now all the trigonometric stuff we uh learned or reviewed in the lecture called trigonometry review so this uh video will not be focusing on trigonometric functions but mainly focusing on the eight functions coming out of the algebra branch and what does it mean to say that I know this type of function that means you know how to answer all of these questions that is you know um you can make a statement about different forms of each type of the function you can discuss how to graph this function you can discuss the domain and range you can discuss the intercepts you can discuss the asymptotes continuity I'm just listing it all I will um provide more details in in a brief moment so let me just finish the list that means you can discuss the monotonicity whether the function is one to one or not whether it's symmetric or not whether it has any special features and you should be able to find an equation of this function given some information and you should be able to know basic applications for each type of the function So eventually you should be able to provide details in all of this um let's call it cells right but remember what I said earlier this is like your knowledge Matrix so everything you learn goes somewhere here right and it should go both ways I can ask you um to identify where does any of the skills that you acquired early belongs in this Matrix or I can ask you to provide more details um for any of this uh intersections um of rows and columns so let me briefly go over um the what does it mean to know the operations Expressions equations Etc and let's start with operations so when we discuss linear function for example we will discuss what kind of operations are there and the idea is that there are altogether eight operations that we identified in the outer range then in trigonometry we identified another 12. and naturally you shouldn't be surprised that those operations kind of turn into functions so you have to know what uh is the predecessor for each original function with all of the consequences uh such as you should know the properties of logarithms for example way before you start discussing logarithmic functions you should know the properties of exponents way before you start discussing uh exponential functions all right next Expressions what is an expression expression is anything that you can write uh using variables constants numbers operations and grouping symbols and you're pretty much allowed to use anything except the equal or inequality sign and again Expressions uh the building blocks in algebra and uh the calculus so there are two tasks that should you should know how to perform generally speaking you should know how to evaluate any expression and you should know how to simplify an expression of course this is where it gets tricky because there are many different types of expression to simplify each expression means certain things that you should uh know now Expressions um I think I already addressed this all right so equations what about the equations equations are on one hand is two mathematical Expressions set equal to one another and what do we do with equations we try to solve them what is a solution solution is any number or collection of numbers that make an equation a true statement what does it mean to solve once again that means we're looking for all such Solutions which we to which we refer to collectively as the solution set now there is a relation between the equation equations or rather the solution set to an equation and the solution set to inequality which we explored in the algebra review video for right now all you need to know is that how to solve each of the different types of the equation so we will discuss that later now generally speaking what is our what's in our toolbox when it comes to solving equations we solve equations by applying the properties of equations some of which are listed here now without knowing functions you may not necessarily know um what are the relations here you may see it as a collection of a number of properties but you also remember that in addition to this properties of equations there are the ones where um for example inequalities why you can add a number to both sides but when you multiply by a negative number you have to flip the inequality so there is more to it than just a number of um Properties or rules uh knowing functions will help us to see much deeper than we are able right now anyway uh let's move on so again remember I'm just going down this list we already talked about operations expressions and equations next I'm going to briefly give you an overview of each of this lines and the first one is a form what exactly are we going to discuss when it comes to a form of a function well by now you should know that functions are examples of relations in which for each input there is only one output and when that's the case a lot of times we can isolate that output in the equation and by doing that we'll create a function in written in its explicit form however it doesn't mean that every function can be written in explicit form uh some of the functions are not and those functions we call implicit functions I don't think we spend too much time in algebra talking about those functions but we will discuss more of that in calculus for sure so for each type of the function you should know what is considered to be the standard form for that function now graphing graphing is easy you just need to know how to graph when it comes to grafting the easiest way to graph is by throwing in put into the function and creating the stable input output and each pair of values of corresponding input outputs creates a point which you plot and by plotting all such points you create what we call the graph of a function now uh you can always use technology but guess what technology actually does this it just does it much faster and then any human would do but as I say here in the notes there is no shame in doing graphs by points so you should be able to graph a function uh given its equation now once you have the graph of the function uh well also by the way for some functions you should be able to graph to produce the graph analytically so for example if and most of the functions that we learn in algebra are such that you should be able to do them analytically meaning without a calculator however there's nothing wrong with just using a calculator it's better than nothing now once you have the graph of a function you should be able to identify its domain and the range so domain of the functions the set of all inputs the range of the function is the set of all outputs and from the graph you should be able to see it just like that now analytically you should know that scenes functions um the predecessor of a function is operations domain is usually obtained by analyzing the restrictions on each operation involved so here is the list of operations and restrictions so you should know that in the logarithm expression the input has to be positive and the base has to be positive We Don't Put negative numbers inside of the even indexed reticle and the base cannot be negative in the exponential expression with the negative exponent because negative exponent means division and we don't divide by zero as we know from arithmetic now operations like addition subtraction multiplication and absolute value do not have any restrictions however trigonometric some of the trigonometric operations do have some of the restrictions and although a lot of them are you appear to be unary uh operations uh some of them can be expressed as the quotients of others I'm referring to tangent being the ratio of sine and cosine so even though it may seem like you just have to memorize the domain of tangent function it actually naturally comes out of the fact that the cosine is in the denominator and cosine cannot be negative so all you have to do is find where it's sorry as a negative I meant to say zero so to find the domain of the tangent all you have to do is to find where cosine is equal to zero now what do we do with all of those restrictions we set them up in this inequality uh it's called compound inequality so we find the domain by solving this compound inequality in general however at this level we we probably never going to see anything too complicated now we're not interested in finding the range analytically uh it's just not what we are interested in and in generally speaking it's not that simple however from a graph you should be able to figure out the range now again given the graph you should be able to find the intercepts there are two types of intercepts x-intercepts and y-intercepts the y-intercepts are the values uh where X is equal to zero so you find y-intercepts by simply evaluating the function now um x-intercepts uh the values where y's are equal to zero but when you set y equal to zero you end up with an equation so remember to find the x-intercepts you really have to be good at algebra for the corresponding types of functions so if you don't know how to solve quadratic equations unfortunately you won't be able to find x-intercepts of a given quadratic function um you can have as little as none of the x-intercepts and you can have as many as infinitely many x-intercepts however a function can have at most one y-intercept does just by the definition of a function remember a function has a single output for each uh input so it is possible for you to have two values uh when X is equal to zero now given the graph of the function you should be able to analyze the asymptotes there are three types of asymptotes and vertical that we express with an equation of a vertical line horizontal that we expressed with an equation of a horizontal line and slant asymptotes which we express with an equation of a of just a line asymptotes could be of higher degree so um certain polynomial functions may behave like parabolas certain rational functions may behave like a hyperbolas I guess uh and the in general asymptote can be non-linear but at this level linear is as far as we'll go all right um given the graph of the function you also have to be able to discuss its monotonicity so monotonicity it's the type of behavior that can be described as increasing or decreasing or constant here's like a proper definition but again looking at the graph it's pretty intuitive when it's increasing when it's decreasing when it's constant and we always go from left to right naturally uh uh and functions that are only increasing or only decreasing are called monotone and if the function is not monotone it may have turning points or it may change the behavior at a vertical asymptote and um why are we interested in monotone functions because um monotone functions are one to one and if a function is not one-to-one it can always be broken down into uh one-to-one components and each component can be analyzed individually why are we interested in one-to-one functions because those are the ones that have inverses and that's a nice property speaking of one to one um once you know that a function is one to one you should know how to find its inverse to find the inverse we follow the steps you swap X and Y in the function and then you solve this resulting equation for y and once you're done you end up with a definition of an inverse function and here's the note that I brought up earlier and basically saying that if a function does not want to one it can be written as a one-to-one a piecewise function and this is one way to define inverse function alternatively we can define an inverse function using a composition of functions uh all right next Once you have a graph you should be able to discuss the symmetries there are two special types of symmetries functions can be called even or odd here's the definition of an odd function and here's the definition of an even function visually it's easy however you should be able to do it uh analytically as well um in case if you don't have a graphing calculator in front of you uh now some graphs have symmetries which are neither old and even some graphs may have axis of symmetries like parabolas that are away from the y-axis have they still have axis of symmetry but we don't call it even nor odd so you should know that for each of the functions now each function has something special about it that's worth uh mentioning uh and you have to know at least one fun fact about each of the function types now frequently you don't have a function until you create it um and to create a function like you have to know how to so for example if the function has been identified by a number of points so you don't have the graph of the function but you do have the um just the points you should be able to figure out the equation of that function most of the time so at least you should know what it takes to find the equation and the rule of thumb is that if in the equation of your function there are n parameters then you need exactly n points that's the rule of the thumb so for example uh if the line straight line has equation f of x equals MX plus b how many parameters are there m and b so you need two points to figure out the equation of that uh line now points is not always what's given sometimes you may be provided some features such as you may not be given a point but you may be given the coordinates or the equation of the um asymptote so in some cases you should know how to use that information to produce the equation now remember we're not learning all this just for fun we're learning functions so we can use them in applications roughly speaking an application is pretty much like any word problem that uses one way or another that function and here's uh step by step approach to solving all the word problems I'm not going to bother you with all the details but you're gonna read the problem maybe several times to understand it you got to introduce the variables you gotta create functions somehow right this is where it comes the skill uh from the previous slide like finding an equation you may have to spend some time finding the equation to create this function now you have to identify the question asked in the problem and related to what you just created somehow so this table kind of provides you a little translation from English to math so for example to find an output for a given input translates into evaluating the function finding the y-intercept means this finding the x-intercept means that finding minimum of the maximum translates into something that would probably spend a lot of time doing in calculus once you identified the question and the solution method you solve the problem and at the very end don't forget to answer the question that was asked in the complete sentence so this is my guide to applications and you should know at least one application for each type of the function some of them are superficial but you still need to know it to summarize uh this is vocabulary that we kind of picked up uh during the pre-calculus uh and it's all related to functions you definitely should practice this vocabulary to make sure we all speak the same uh language this concludes my review of the I guess it's not the review itself but it's a preview because next I'm going to create a review for each type of the functions other than trigonometric ones because I discussed those already in the video called trigonometry review so next I'm going to create a sequence of eight videos one for each type of the function if you have any questions feel free to reach out to me