 Hi, I'm Zor. Welcome to Unisor Education. I would like to continue talking about general aspects of functions, what function actually is, what kind of functions you can deal with, and specifically, today's topic is monotonic functions. So very, very briefly, monotonic means it's basically either increasing or decreasing. It's kind of natural understanding of what this function is about. But let's talk about what it means, increasing or decreasing. First of all, if this is a function, it means there is a domain for this function, where the function is defined, and there is a range, there is a co-domain. Now, let's ask a question. If I have a function which for each flower puts into the correspondence the color, can we talk about increasing or decreasing of this particular function? Obviously not. So what does it mean? The function is increasing or decreasing. Well, first of all, it means that in the domain, there must be something which we can talk about as a relationship less than or greater than or equal, etc. Now, the most typical example of this domain is real numbers. So basically, in the mathematics we will be dealing with, mostly we will deal with the functions which are defined on the domain of the real numbers. Among real numbers, we do have this relationship. We can say that one particular real number is less than another. Okay. Now, since we are talking about increasing or decreasing, the same kind of relationship must exist among the various of the function. Now, the various belong to the co-domain. They are in the range of the function. So wherever the function takes the values from, also should be the same kind of relationship. So we can always talk about two particular values of the function. Function of one should be less than or greater or equal, whatever, than the value of the function at another point, at another value of the argument. So to define increasing or decreasing functions, we have to have a relationship like this one. Less than or greater than or whatever, among the arguments and among the values of the function. And again, the most typical example of such a set which has this type of relationship is real numbers. We can always talk about two real numbers as one being less than another. So to make the long story short, we are considering right now when we are talking about monotonic functions. We are talking only about the functions which are defined on the domain of the real numbers and take the values real numbers. Okay, that's done. No flowers, no colors. We are talking about numbers, real numbers, two real numbers. Now, let's just put together definition of the monotonic function. So the monotonic function is the following. If you have two different arguments, u and v, both belong to the real numbers, both are real numbers, then the value of the function in these two arguments has exactly the same relationship. So if the function is defined at two different points, real numbers, u and v, and u is less than v, monotonic function has this property. The value of the function in the lower, in the smaller number is less than the value of the function in the bigger number. Now, does it mean that the function is supposed to be defined for all real numbers? Not at all. There are some monotonic functions which are defined only on the very narrow interval or something like this. Not necessarily entire set of real numbers. Same as values. Not necessarily the function values are covering all the real numbers, it can be a subset. But in any case, wherever it's defined, if we pick two different points from the domain, then the values of the function must comply with this particular inequality. Now, let me ask you this question. Does it mean that this results in this? So if the function is monotonic, so for any two numbers, one is less than another, the values of the function are in exactly the same relationship, does it mean that if the value of one is greater than another, then the value of the function would be greater in this point than in this point? Absolutely yes, because u greater than v means v, this is exactly the same, it's equivalent to v less than u, right? Now from this, using the property of monotonic function, we can say that f would be less than f of u. And this is exactly the same as this. So if one value is less than another, then the other is greater than this one. So basically, if function is monotonic, then not only less than is preserved, but also greater than is preserved. Finally, how about this? What if we will take values which are exactly the same? Well, then obviously we will have to have the same value of the function just because the function always have one and only one value for some concrete value of argument. So these relationships are preserved always. This is just because it's a function, and these two just because it's a monotonic function. Now, what does it actually mean as far as relationship between domain and the range? Well, the very basic and very fundamental and extremely important property is that monotonic function establishes something which we call one-to-one correspondence. So domain and the range are in one-to-one correspondence relationship. What does it mean? It means for any one value of the argument, we can find the value among the range, and for any value in the range, we can find one and only one. That's very important, value of the argument. So the first one that for any point in the domain exists something in the range, that's obvious, because this is exactly the function of a relationship. That's the definition of the function for every element from the domain. From every element from the domain, there is some kind of element in the range. But now we're talking about the inverse relationship. If it's a range, it means it's value of something. So my most important question is, is it the value of some one particular argument or two arguments actually match the same value among the range? Now, that is impossible. For a monotonic function, we cannot have two different arguments which are pointing to the same value of the function. Why? Because of this. If these two arguments are different, then one is less than another, which means that the corresponding value of the function should be less, not equal, less than another. So if we will take any element from the range, which means it's the result of the function applied to something. So that something should be unique. It cannot be the same value. This is impossible for a monotonic function just because the monotonicity of the function itself. If arguments are different, then because of this, function values must be different, which means for every single point, every single element in the range, there is one and only one element in the domain function of which is equal to that element in the range. And that's what it actually means one-to-one correspondence. For every element in the domain, X, we can find the corresponding value of the function just applying the function itself. But then that particular element from the range can correspond to only this particular value of the function it came from. There is no other because other would be less or greater and the function value will not be the same. So one-to-one correspondence, that's extremely important for monotonic functions. And one of the consequences of this one-to-one correspondence is existence of the inverse function. So if you have the main to range with a function F, so X goes to F of X. Now, there is an inverse function which always finds what exactly was the element in the domain, the result of which is a particular element in the range. So we have the function which goes this direction, which we will call G. And this function puts into the correspondence for every element in the range, some element in the domain, which actually a prototype from which this particular element was obtained using the function F. So we were talking about the inverse function before, but this is a perfect example of the case when the inverse function definitely exists. So for any monotonic monolithic, for every monotonic function, there is always inverse function. And since we are talking about real values for the main and the range, we are talking about only the function which is defined on the domain of real numbers and take values, the range is also among the real numbers. So both functions F and G are both real values to real values functions. Okay, now, well, probably would be nice to have some kind of example. Well, let's have a very simple example. Let's put the function, something like this. So for every element X, real value, we take one-third of it and that's the value of the function. Now, what is the reverse function? Let's just think about it. If we take, for instance, one, this is X, this is Y, we will have one-third. We have three, we will have one. We have 27, we have nine. So this is the function which is our original monotonic function. It's obvious to monotonic when X is increasing, Y is increasing as well. Now, what is an inverse function? Well, the inverse function would put into correspondence to one-third, one. To one, it would put into correspondence three, to nine, 27. So obviously the inverse function would be this. And it's also monotonic. It's also increasing, because obviously this is true. And this, from this being true, we basically follow that this is true since it's one-to-one correspondence. It means that the reverse function is also monotonic. So, but now these two functions are inverse to each other. Both establish one-to-one correspondence between all the real numbers, in this case, the main is all the real numbers, and the range is also all the real numbers. And this is just one of the examples. Another example would be, for instance, just give you another example. One function would be Y is equal to a cube root of X, and inverse function would be X cubed. Now, let's just think about it. If we will start with some domain element, for instance, 27. Now, if we will apply this function, we will get three, right? The cube root of 27 is three. Now, if we will apply this function now, three to the power of three would be, again, 27. So, direct function maps number to another number, and inverse function maps this function back to the original. And that's the purpose, basically, of having inverse function. So, these two functions are inverse to each other as well. So, we're talking about one-to-one correspondence. We were talking about inverse function. Here's an interesting comment, which I actually planned, this is my plan, by the way, I planned to mention. So, we know that every monotonic function establishes a one-to-one correspondence between domain and a range, and there is an inverse function. Now, here's my question. Is opposite true? So, if, for instance, we have a function which has inverse function, a function which establishes one-to-one correspondence, between domain and a range, does it mean it's monotonic? The answer is no. And to prove it, let me just make the following example. I will use the graph, although we didn't really talk about graphs much, but probably you understand what graph actually means. And here is the graph of the function. I will start with the graph and then we'll derive the formula. So, let's say I'm defining the function only on this particular interval from minus one-to-one. And the way how I define it is the following. Here, I will define it this way, but here I define it this way. I'll put a little arrow here, which means that a point zero function is equal to zero according to this graph. So, this is x, this is y. So, for every x to find the value of the function, I have to draw a perpendicular and this is the corresponding y. Now, this is from minus one-to-one domain of the function. Now, this is, let me just make it a little better here. Don't need this anymore. Okay, this is the range of the function from zero to two. So, the function is defined from minus one-to-one and for each value of x to find the value of the function, we will draw a perpendicular to the x-axis, to the intersection with my graph and then to the right or to the left to get the value of the y. Now, is it the function which defines one-to-one correspondence? Oh yes, absolutely. Pick any point here, you will get the point on the y. Pick any point on the y, y-axis, let's say this point and then draw the horizontal line until it intersects the graph, in this case, at this point and draw down the perpendicular to x. Now, if it's something else, let's say this is y, then you draw the line parallel to x until it intersects the graph and go down. It's always one particular point, wherever we pick the y, we will always, drawing a horizontal line, we will always intersect our graph in one and only one point, either on this piece or on that piece. So, one-to-one correspondence is established. Now, how about the formula? The formula is y is equal to, on this particular, it's equal to x, this is x, this is y, and they have the same value, if x greater or equal than zero. And this is minus x plus one, minus x plus one for all x less than zero. So, this is the formula which represents our function. It's a formula, it's two different formulas, but it doesn't really matter because these intervals are different. So, the whole interval will be covered by these two inequalities. So, the function is defined. For any x, I can find the value of the function. Okay, if x is greater or equal to zero, I use this function, this formula and for negative x, I will use this formula. So, I will always find the value of the y. Now, can I find the value of the y? Well, obviously yes, because if my value of the y is below one, I will use this formula and if it's above one, I will use this formula. So, basically the inverse function would be y is equal to x again if x is greater or equal to zero. And this function, how can I resolve that would be one minus x, right? One minus y, one minus x. If x is less than zero. Now, in both cases, x belongs to minus one, one interval. So, the function is defined only within this domain. Or I can put this restriction here, greater than zero or and less than one and here less than zero but greater than minus one. That's the same thing. All right, so these two formulas are basically describing my inverse function. So, inverse function exists in many cases, not necessarily in the case of monotonic function because this function is obviously not monotonic. It's decreasing here and increasing there, right? So, this is the relationship. Any monotonic function is inversible. Not every inversible function is monotonic. There are inversible functions which are not monotonic. Okay, that's fine. And now there is an interesting aspect of inverse function which is more about inverse than monotonic but obviously since every monotonic is inversible so it's true for monotonic function as well. Let's talk again about the graph of the function. Let's talk about monotonic function and their graph. So, again, let's draw our system of coordinates and let's think about some monotonic function. Just as an example, this function. Doesn't matter what's the formula behind it. What is important is that this function as we see is monotonic, right? Question is, how will the graph of inverse function look like? That's very interesting. Now, let's think about it this way. Let's say this is the graph of the function y is equal to f of x. Now, if there is one particular point on that graph which has coordinates a, b. Now, what does it mean? It means that if I will draw down, I will get to the a and if I will draw across, I will get to the point b. That's what it means. Now, again, it's a little bit ahead of all my discussions about graph, but if you want, you can go there and then return back to this because it's important property of the inverse, inversible and therefore monotonic functions as well. So, if the function belongs to a graph, it means that projections on both co-ordinate axes are correspondingly a and b. Now, from another standpoint, since this point belongs to the function y is equal to f of x, what does it mean? It means that b is equal to f of a, right? So, if I will use the a as an argument, I will get the b. That's what actually means that this particular point belongs to the graph of this function, right? Now, what is inverse function? Inverse function is the function which, if you take b as an argument, you will get a as the value, right? So, direct function takes a as argument and gets b as an argument. Inverse function takes b as an argument, it gets a. So, if my function is called g inverse function, if I will take b as an argument, I will get a. Now, from the graphical perspective, it means that if I would like to draw graph of the g function, it means that function with, the function g being applied to the point b as an argument, b as an argument, will get a as a value. This is point a b, sorry, b a, right? So, if point a b belongs to this graph, point b a belongs to this graph. So, always, if there is a one point which has coordinates a and b on the graph of the function y is equal to f of x, and there is an inverse function which can be designated as y is equal to g of x, if we will put b as an argument, we will get s a as a value and the point b comma a. So, x coordinate will be b and y coordinate will be a. This function belongs to the inverse function, this point. But now, let's talk about these two points. Points a b and b a, how do they position one relative to another? Well, it's actually very easy. Let's draw a bisector of this angle. So, this is 90 degree, this is 45 degrees. Now, where is point a b? Now, very easy statement in this case is that these two points are symmetrical relative to the bisector. Now, it's very easy to prove genetically, but since this is not a geometry lesson, I will probably refer you to the whole chapter called geometry in this course. You can try to basically prove it yourself, but believe me, it's very easy to prove that these two points are symmetrical relative to the main angle bisector, which means basically that the whole graph of inverse function would be symmetrical relative to the bisector. So, let me again draw, let me try maybe a different color, draw the original graph, which looks like this approximately. And it goes through, okay, something like this, right? This is original. Now, let me draw a graph which is symmetrical to this one. Relative to this bisector. Well, it probably looks like this. So, these two graphs are symmetrical relative to the angle bisector. This is the original monotonic function, and this is an inverse function. So, inverse function always symmetrical with the original function relative to the angle bisector. Well, that's actually all the general properties of the monotonic functions, which I wanted to talk about today. As usually, I do encourage you to register as a student, have somebody, or maybe yourself, if you want and register as your supervisor or parent, maybe, go to Unisor.com and try to take the complete course with all the lectures, exercises, and exams, which is very, very, very important. Good luck, thank you very much.