 I'm Zor. Welcome to Unisor Education. Today's topic will be functions. There are different definitions, certain principles which I would like to explain today, very briefly, because it's a really very American class subject. Okay, what is a function? Philosophically, generally speaking, function is certain dependency between something and something. So, to be a little bit more precise, let's imagine that you have two sets, absolutely abstract sets, one sets, one set and another set. We will call this one domain and this one we will call codermain. Let's imagine that you have certain rule which, given an element, any element from the domain set, you will be able, by applying this rule, come to a certain element of a codermain set. Well, basically this is a definition of a function, function of dependency. So, this is a definition of a function which is defined on a set of elements which we call domain and it takes the values in the set which we call codermain. Not necessarily all the values will fill out completely the codermain. So, maybe all the images of elements of the domain will actually be a subset of the codermain set, in which case we will call this subset of all the images of the elements of the domain, we usually call it at range. So, the function, if we want to define it a little bit more precisely, should be defined as the combination of one set, which we call domain, from which we will take the elements which are called arguments. So, these are elements of the domain which we call arguments. And then we have to have another set which we will call codermain and then we have a rule which allows us for any element from the domain for any argument to be able to find the corresponding value, this is called a value, an element in the codermain. And a range is a set of all the different values which this particular function which we can call f whatever, which this particular function can take. So, this is not really very mathematically rigorous definition but still kind of an explanation which allows to understand what the function actually is. Usually in school people are talking about functions which are defined on the set of all the numbers. So, the domain is usually the numbers and the values, the codermain is also usually the numbers but in theory function can be any. Let me give you just a very simple example. You go to a forest and you see flowers, every flower has a color. Well, what is the function in this particular case is a set of all flowers and the codermain is a set of all colors. Well, obviously not necessarily the color of all the flowers you see somewhere in the forest or wherever you go would fill up completely all the spectrum of all the colors. So, the range can be a little bit smaller. For instance, you see only red flowers. Then the range will be all the variations of red and the blue ones are not really in the range. So, the codermain is actually all the colors like blue and red and yellow and whatever but the range is actually only different shades of red. For instance, it can be different. Okay, what other functions can be defined in this particular set? Well, we can have a different function which is defined in the same domain which means elements of this domain. Flowers will still be arguments but the function itself will be let's say a size of a flower. Whatever the size might be, I mean bigger, smaller, measured in millimeters, centimeters, inches, doesn't matter. So, in this case again domain is a set of flowers and what is our codermain? Well, that's basically all the different numbers. If you measure in let's say centimeters, then from tiny flowers of fractions of centimeters to maybe gigantic flowers, I think the biggest flower can be like at a meter, I don't know, two meters. No more than that probably. So, this will be the range. So, the codermain will be all the numbers but the range, all the real sizes which flowers can really have is definitely smaller than this. Let's say it's numbers from zero to 200 centimeters. There are actually some other functions which might be defined on the set of flowers like shape of leaves and all the kind of shapes can be that all the different kinds of shapes can definitely be defined here and where this flower is growing, for instance, the continent or the temperature this flower is usually comfortable with. So, there are many different characteristics which we can talk about as functional dependencies between the flower and something. And now we are approaching a very interesting point. If I will tell you only that the color of a flower is let's say red, does it define the flower itself? No, there are many different flowers which are red. Okay, what if I will tell you? Okay, the color is red and the size is such and such and the shape of leaves is such and such and the temperature and the continent and the form of roots and whatever other different components I can talk about each one of them is a separate function. This is one function, this is another function, this is a third function, etc. All the different characteristics each one of them representing a function defined on the set of flowers. Does it really define completely what was there? Well, as you see we have to really go from value to the argument of the function or maybe value of many different functions defined on the same argument and try to determine what was the argument which produced these values. So, basically as you see we have a different concept here. We have a concept of inverse function. So, what is this? Again, if you have certain domain these are elements and you have certain co-domain these are elements and you have some kind of correspondence from each element you go and you find the value of this argument among the different elements of the co-domain. If you can define absolutely deterministically knowing the value you can actually go back and find the argument. Well, what is this? This is another function actually which has co-domain as an argument and domain as the value. So, one function is defined from here to there this is argument this is value and another function is from the value back to the argument. Is it always possible to find this other function which is called by the way inverse? Well, not necessarily. If I have something like edges wrong here then yes it's obvious that for each element of the co-domain I can find by following this line backwards I can find what the argument was and that would be the value of the inverse function but what if situation is like this what if the same element of the co-domain the same value can be derived by applying the function to two different elements of the domain. In our example with a flower what if there are two different flowers which have exactly the same color as you see color does not define flower completely. Okay, so inverse function is not always defined properly or I mean it's not defined at all in certain cases but in some cases it is actually defined obviously in school mathematics we usually deal with functions which are which which have domain and numeric or all the different numbers and the co-domain and range of the function always the numbers and among these numerical functions there are many which really have something like one-to-one correspondence between the arguments and the values so you can always go backwards from the value back to the function to find what was exactly the inverse function well typical example is if function is described as a formula when this is an argument and this is the value and you have to just multiply the argument by itself three times to the power of three then obviously there is a inverse function which basically does the same thing if you take number two this function converts two to eight and this function converts eight back to two all right so maybe I'm jumping a little bit ahead I didn't really talk about powers and the roots etc but in general I'm sure people who went through normal school mathematics now what I'm talking about here so in this particular case function y equals x to the third degree is reversible or inversible it has an inverse function and this is the one but let's consider a different function what if I will have a square here this is a typical example of a case like this because as you understand two goes to four and minus two square also is four there is no such thing as a function which will convert one number four into two two and minus two well conditionally you can say something like y is equal plus minus square root of x but this is not a function these are actually two functions one function which returns the positive number and another returns a negative number so this is just a conditional expression it's not really the function itself okay so as you see functions can be reversible or reversible but inverse function does exist and that might not be the case now let's consider again a function which is described in this particular picture and we will introduce two new concepts the concept of restriction and expansion or extension what is a restriction and extension it's actually a very simple thing let's consider you have a function which represented exactly by this picture so there are four different elements in the domain and no more just these four and you have three different elements which actually constitute the co-domain now if the function is defined like this that's our initial function now let me define a different function function which is basically derived from this one by restricting the arguments only to a subset of the domain so the function now a new function is defined on the subset of the main and obviously the values will be also members of the co-domain elements of the co-domain but it will be probably a smaller range all right so this is a restriction so we restrict it we restrict our function to a subset of the domain so a new function now why is it a new function because we have a completely different domain and completely different co-domain because the subset now of the original domain which is a different set is a domain of a new function so if it has a new domain it's a new function but this new function definitely is related to the original because the domain is a subset now what if we have the other way around so instead of restricting our function to a subset of a domain let's assume that initially we have only the function which is defined on the subset and the values are here if it's defined initially original function and now we have complemented this function we have expanded it we expand we expand the function towards a bigger domain so if function was defined here and these are the values but now they are considering a new function which is defined from the bigger set which is a superset of our original set and obviously a new range will be a superset of our original range now this function is called extension just a terminology no video actually about it all right what else about the functions quite frankly if function is not numerical which most of functions which we are studying in mathematics the function is not numerical there's not really too much we can talk about it's more like philosophy and and other subjects in mathematics we usually are studying numerical functions functions which are defined somewhere among the numbers and take value among the numbers so what are the ways we can represent this numerical function well actually there are many different ways and one of them I have already just exemplified like y equals x where or something like this this is the formula so one thing is to represent numerical function is formula here's an interesting point formula is something which allows to describe the function quite generally actually if you have a formula y equals to f square well does it define the function well yes and no in general we can say yes because well just take any number square it and you will get another number so from argument you get to the value of this function using certain rules of algebra and more precisely you really have to talk about domain and and and co-domain or range of the function what can be squared if I will tell you that let's consider the function which is defined only on a set of natural numbers one two three four five etc so if domain is natural numbers what will be the range well range also will be natural numbers but all of them no not all of them obviously the range is something like one four nine 16 etc squares so if you have the main one two three four etc then your range will be one four nine 16 etc so basically you have to specify not only the formula but also what's the domain range is not really necessary to specify because you can always find out what's the range by substituting these numbers into the formula so this function which is defined on a set of natural numbers is different from the function which is described by the same formula but which is defined on a set of let's say rational numbers why because different domain if it's a different domain it's a different function function is the combination of three things the domain the co-domain and the rule which corresponds to the main point or element to the to the co-domain however we usually omit this type of discussion in in our everyday mathematical life because it's probably not really very important it's not very interesting formula is really sufficient because you can always think about the function which is defined only on numerical but only on natural numbers as a restriction which means defined on the smaller subset of another function which is defined on the set of rational numbers and that in turn is a restriction of another function which is expressed by the same formula where domain is the set of all real numbers and if you're really sophisticated and you know what the complex numbers are you can always say that the function defined by this formula on a domain of real numbers is a restriction of another function which is exactly the same expressed by the same formula but defined on a set of complex numbers every new function which we are said which we are talking about is an expansion of the previous one so we can always say okay this function is defined in the biggest domain we can possibly can and in reality it's either complex numbers or if you're okay with real numbers then that's the real numbers which is the biggest possible domain people really are interested in so there is no really need to talk about the main itself because it's always restriction or expansion of another function which is basically the same thing the same formula so the formula is one thing which we can talk about as a representation of a numerical function well what other forms well obviously there is a very simple form a table you can say that this is the main this is the main this is called the main and you can say that the main is numbers 1.5 in which case the function is 3 the main can be 4.2 in which the function is minus 2 and element is 7.57 and the value of the main is 0.66 okay is this the function yes absolutely the main is a set of three numbers 1.5 4.2 and 7.57 and the range which belongs to the common main of all the different numbers the range is also only three numbers which is 3 minus 2 and 0.66 yes this is also a function obviously the representation of a function in this particular way is definitely restricted because you can't put like infinite number of arguments getting a certain number of values of this function however for certain purely practical tasks this is the only way we can represent the function let's assume that you are measuring temperature in the city of new york during 1957 so you will have 365 days and for each day you have certain temperature which you can write as sorry as a temperature here and the date itself will be on the left so here we will have a set of dates and here we will have a set of temperatures the function is defined for every day we have a temperature that's a function it's represented by by a table and by the way it's not even a numerical function in the full sense of this word because the definition the domain is not a number it's a date however the the value is a number so it's half of the numerical function if you wish the domain is not and the code domain is not that is a number but anyway table is a respectable way to define the function and in practical examples it's really quite common let's go back to mathematics from the practical life mathematicians don't really master's practice they usually try to stay within the framework of their theory okay what other ways to represent numerical function in this case well you probably have heard about the graph graph is also a representation of a functional dependency let's talk about certain function for instance this function well I'm sure everybody remembers Parapola so this is a graphical representation of this function how can we use this particular graph very simply if you have number one here you go up perpendicularly and then go to the left and you found one if you go here two you go up and then left and you find four so for every number if you want to know what's the square of this number what's the value of this function you actually find a point on the x-axis which corresponds to the argument then go up perpendicularly go left and find what's the value on the y-axis so this is the rule for every point on the x-axis which represents a number obviously you can find using this procedure what's the value of the function which means the function is defined that's it all right so that's all about the graphical representation which we will definitely spend a little bit more time some other day just drawing different graphs for different functions so you have at least these three different representations which mathematicians might use for describing this functional dependency between domain and co-domain between argument and value here is another interesting way to consider functions which is maybe a little bit higher in the mathematical level than usually people learn in school however it's a very simple concept so just try to explain it here so let's say you have certain domain here and you have a function which for every argument gives you some kind of value of this function so this is the main and this is co-domain all right great now let's consider a different function let's call this function f let's consider different function for instance we have another function which has this set which is a co-domain for the first function f as a domain and then goes further into a different so this now becomes a domain for function g and this becomes co-domain for it so from this element of the original domain oh let's forget about the domains and co-domains let's call it a b and c that would be easier so there is a function f which is defined as a function from a to b so every element of a has certain image certain value in b and then there is a function g which further makes the correspondence between every element of the set b to a set to an element of set c so what it actually means is that for every element a which belongs to the set a i can actually find element b in the set b using the function f and from this element b using the function g i can go to an element c so we are consecutively can apply functions b and then g to an element a to get to the c what it actually defines is this because for every element a i have an exact rule how to get to a set c to element c what's the rule well first you apply a function f and then i will use a plus in the circle as a combination then as a continuation or a combination with the function f you should apply a function g now if everything every a b and c are let's say sets of real numbers then we can talk about something like this c is equal to function g applied on the result of function f applied on a that's what it is so we take the a first we use the function f to get to this but both of them are actually numerical so everything from number to number and then use this number as an argument apply function g to get to get to the c now this is called the combination of these two functions called the composition in theory it it is actually very much close to multiplication because you're consecutively applying one after another is you're multiplying one number by another and then by another so if you consider a set of all the different functions which are defined all of them are defined let's say on a set of real numbers then you can talk about this multiplication of the functions it's a little bit higher concept and it relates to functional analysis etc which people are usually learning in colleges but in theory you understand that there is something which is a composition of two functions which really gives you another function so it's kind of an operation between two functions two functions can be really operated upon like two numbers can be multiplied and the third number you will get as a result two functions if they're properly defined like one of them is having the range where the second one has a domain if they're properly defined then you can consider the operation of composition in a way similar to multiplication of numbers so that's another concept which I wanted to to talk about so basically that completes the general introduction to what functions actually are and what we will do in in algebra for instance that's what people do in algebra they're learning different functions and how they are really represented by using formulas graphs what kind of functions exist etc so you can say that maybe the whole algebra is a subject is studying the different functions well that's it for today thank you very much