 Welcome back to our lecture series math 1050 college algebra for students at Southern Utah University As usual, I'll be your professor today. Dr. Andrew Missildine and lecture 35 We're almost to the end of our unit five about algebraic functions So what does one mean by an algebraic function? We basically mean all of those functions that we can build using addition, subtraction, multiplication and division and also exponents as well And so this one include the polynomial monomial power functions We've talked about already rational functions through thrown in that division there But we also have to incorporate their inverse functions, which we talked about inverse functions much earlier in this series This lecture in many ways will be a review of what we've learned about inverse functions before But we're going to use that as motivation to introduce radical functions So we're talking about things like square roots and cube roots and fourth roots and the like right Which we'll see in a moment that we can represent each of these radicals as in fact exponents and Radicals are really just going to be the inverse functions to monomial functions, which we studied earlier in chapter four here So to talk about inverse functions, there's a few things we have to remember So remember that we say a function is one to one Well a function is one to one when its graph passes the horizontal line test So for example, if we were to sketch a graph real quick So maybe we have like our x and y axes right here and let's draw one to one function. Maybe something like this Seems like a good picture and the point is whenever you draw a horizontal line It'll intersect the graph at most one place And so this is an example of a function passing the horizontal line test This is in contrast to like say your standard parabola Right which the parabola Fails the vertical line to a horizontal line test excuse me because there can be more than one place where the horizontal line hits the graph Now there is a way of sort of fixing this we'll see this a little bit later in this series in this lecture I should say here, but that's what it meant for a function of one to one Now what we learned previously is that if a function is one to one then it has an inverse function a Function which undoes the original function So you have your function f if you compose it with f inverse You'll have the property that this is just the same thing as x itself the identity function that f inverse after f The net effect is as if nothing happened whatsoever a very benign and simple example will be like if y equals x plus 2 and you have y equals x minus 2 when you put these things together When you compose these functions as if nothing happens right if you take x plus 2 and then you subtract 2 from it You just end up with x So addition subtraction these inverse operations multiplication and division are these inverse operations if I take 2x and then I divide it by 2 It's as if nothing happened right the net effect is is nothing and so what's the inverse operation when it comes to powers? So power functions which we saw at the very beginning of chapter 4 And then we generalize these as we went through polynomial and rational functions If you have something like x cubed, how do you get rid of the x cubed power? And we we've seen this already how to do this, but we want to make it much more formal in this conversation So imagine we have some type of monomial function like f of x equals x to the n or in this case We'll take x cubed and let's focus on them at the moment on the odd case All right, so n is an odd integer It could be positive like a monomial or it could be negative like the reciprocal functions We've seen before and let's just restrict our attention to the non-negative domain for a little bit zero to infinity this thing is defined For negatives as well But the main reason we want to restrict our domain from zero to infinity here is so that we can come in harmony With also the power functions where x was an even power like x squared x to the fourth x to the Negative two things like that then in this situation when we restrict our function When we restrict our function to just the domain zero to infinity both the odd case and the even case Will be invertible that as they'll be one-to-one functions. Let's take a look at that graph for a moment So if we were to again graph our x-axis and y-axis, I know I just did them backwards there I know the difference if we were to graph the function y equals x cubed This would be the standard graph, but if we just want to focus on the positive domain, we get something like this Okay, y equals x to the n like so and the reason here is that even if it was like a parabola, right? The problem the left-hand side looks a little bit different But if we ignore the left-hand side all of these functions look about the same and the bigger you make in the steeper This graph becomes right here You'll see that on this domain zero to infinity our function is in fact one-to-one it passes the horizontal line test Therefore if we were to reflect this graph Excuse me if we reflect this graph across the diagonal line y equals x We would end up with a curve that looks something like the following Here in green and this would be what we call y equals the nth root of x or in terms of exponences is denoted as x to the one over n power and So why do we use this reciprocal exponent right here? Well, the idea comes from exponent laws that if we had something like x cubed We want to raise it to a power say a so that this gives us back x But by exponent rules if you take three x to the three to the a this is the same thing as three to the a Which on the right-hand side x by itself is just x to the first This would have to mean that three a equals one that is a equals one-third Let me say that again if we take x cubed and raise it to the one-third power Then by exponent rules you multiply those together you get x to the three-thirds power Which is x to the first which is equal just to x itself So the way to undo a monomial is to take these reciprocal powers And so this nth root this nth root of x right here the nth root of x is defined to be the unique Positive number so that when you take it to the nth power you get back in x so radicals which also coincide with Rational exponents here radicals are constructed to be the inverse operations to the monomial functions. We introduced previously now in terms of domains and Domains and ranges of radical functions. Well, it does depend on whether we're even or odd in That conversation so I do need to make a few comments about that just right here right now So imagine we have some odd power So you have y equals x to the n and n is equal to some odd number So what that tells us is that well the domain of this thing is Going to be all real numbers the range of this function is also going to be all real numbers And then the graph if we were to sketch it real quick the graph of our function on This domain You know, it's basically going to look something like the following Again, the exact steepness would change based upon the power But you're gonna get a graph to look something like this which you can see that this graph does pass the horizontal line test So this function is one to one in which case then its inverse function would look like y equals the nth root The nth root of x or x to the 1 over n where again n is still an odd number In that situation you would see that it's domain One thing to remember out inverse functions is that inverse functions switch the roles of x and y So the domain becomes the range and the range becomes the domain So for these odd powers these odd radicals your domain would be all real numbers and then your range would be all real numbers In which case if you then reflect this graph across the diagonal You're gonna get a graph That looks something like the following y equals the nth root of x and this happens when n is an odd number of course In contrast If we were to do say y equals x to the n and we take n this time to be not an odd number We just did that one an even number In this situation, I'm going to restrict the domain To be zero to infinity like we saw before Okay, and then the range likewise is going to be Zero to infinity now. Let's think about the graph of this thing just for one second before we continue on with our picture here So if we just look at the left-hand side at the right-hand side of this graph again You give this picture like we saw above now if you took the whole domain You would be getting this reflection on the other side. This thing is going to be symmetric with respect to the y axis It's an even function after all this is why we call them even functions So this graph if you look at the the whole graph It's not one-to-one because of the violations we see right here So if we remove the left-hand side this violation disappears and so if we just look at the right-hand side this would then be a one-to-one function and This is the function then we invert For which we get this graph like we saw before For which we then are going to get y equals this nth root of x right here And so some important things to mention if we take if we take y to equal the nth root of x when n is an even number Then the domain and range are going to swap locations here Just like they did with the odd case and so we end up with the domain Equaling zero to infinity the range is likewise Gonna equal zero to infinity and this is an important difference We should pay attention to when we talk about even radicals versus odd radicals for example You'll notice that the domain lacks any negative numbers which agrees with what we've seen so far because if we take f of x to Equal the square root of x and then we look at f of negative one that would be the square root of negative one Which is the imaginary unit i right in terms of graphs We want to follow the domain convention that we take only real numbers in and only real numbers out So we have to restrict the domain so that the output is a well-defined real number That's why we have to kind of restrict this going in over here Likewise, we have to also restrict the domain or the range because we can only get positive numbers coming out of this thing now because we did Because we did take away half of the graph right so you take like your parabola right here Half of the graph is missing this is going to affect things as you try to solve equations involving radicals Like if you have the equation x squared equals 2 we'll do a perfect square of x squared equals 4, right? The issue you have to be aware of here is that when you take the square root on both sides to solve this equation You end up with x equals plus or minus 2 right? You have to consider the two possibilities because there's the possibility on the right There's also the possibility on the left which the inverse function doesn't look at that Because to find an inverse function we do need a one-to-one function And so in this lecture, we're going to talk some more about these these square roots these cubits these radicals This video serve as an introduction. We'll talk about the things we like to talk about with functions here We're going to talk about their graphs in just a second. We'll talk about solving equations involving radicals And we'll also talk about solving inequalities involving radicals as well